P. 54 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fconst3m 5301*	Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B}))))
Theorem	fconst4m 5302*	Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A)))
Theorem	resfunexg 5303	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) ∈ V)
Theorem	fnex 5304	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5303. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹 Fn A ∧ A ∈ B) → 𝐹 ∈ V)
Theorem	funex 5305	If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5304. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ B) → 𝐹 ∈ V)
Theorem	opabex 5306*	Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
⊢ A ∈ V    &   ⊢ (x ∈ A → ∃*yφ)    ⇒   ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ φ)} ∈ V
Theorem	mptexg 5307*	If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (A ∈ 𝑉 → (x ∈ A ↦ B) ∈ V)
Theorem	mptex 5308*	If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
⊢ A ∈ V    ⇒   ⊢ (x ∈ A ↦ B) ∈ V
Theorem	fex 5309	If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
⊢ ((𝐹:A⟶B ∧ A ∈ 𝐶) → 𝐹 ∈ V)
Theorem	eufnfv 5310*	A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ∃!f(f Fn A ∧ ∀x ∈ A (f‘x) = B)
Theorem	funfvima 5311	A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
⊢ ((Fun 𝐹 ∧ B ∈ dom 𝐹) → (B ∈ A → (𝐹‘B) ∈ (𝐹 “ A)))
Theorem	funfvima2 5312	A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
⊢ ((Fun 𝐹 ∧ A ⊆ dom 𝐹) → (B ∈ A → (𝐹‘B) ∈ (𝐹 “ A)))
Theorem	funfvima3 5313	A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (A ∈ dom 𝐹 → (𝐹‘A) ∈ (𝐺 “ {A})))
Theorem	fnfvima 5314	The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
⊢ ((𝐹 Fn A ∧ 𝑆 ⊆ A ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))
Theorem	rexima 5315*	Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ (x = (𝐹‘y) → (φ ↔ ψ))    ⇒   ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ))
Theorem	ralima 5316*	Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ (x = (𝐹‘y) → (φ ↔ ψ))    ⇒   ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∀x ∈ (𝐹 “ B)φ ↔ ∀y ∈ B ψ))
Theorem	idref 5317*	TODO: This is the same as issref 4630 (which has a much longer proof). Should we replace issref 4630 with this one? - NM 9-May-2016. Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)
⊢ (( I ↾ A) ⊆ 𝑅 ↔ ∀x ∈ A x𝑅x)
Theorem	elabrex 5318*	Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
⊢ B ∈ V    ⇒   ⊢ (x ∈ A → B ∈ {y ∣ ∃x ∈ A y = B})
Theorem	abrexco 5319*	Composition of two image maps 𝐶(y) and B(w). (Contributed by NM, 27-May-2013.)
⊢ B ∈ V    &   ⊢ (y = B → 𝐶 = 𝐷)    ⇒   ⊢ {x ∣ ∃y ∈ {z ∣ ∃w ∈ A z = B}x = 𝐶} = {x ∣ ∃w ∈ A x = 𝐷}
Theorem	imaiun 5320*	The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (A “ ∪ x ∈ B 𝐶) = ∪ x ∈ B (A “ 𝐶)
Theorem	imauni 5321*	The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
⊢ (A “ ∪ B) = ∪ x ∈ B (A “ x)
Theorem	fniunfv 5322*	The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹)
Theorem	funiunfvdm 5323*	The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5322. (Contributed by Jim Kingdon, 10-Jan-2019.)
⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))
Theorem	funiunfvdmf 5324*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5323 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
⊢ Ⅎx𝐹    ⇒   ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))
Theorem	eluniimadm 5325*	Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
⊢ (𝐹 Fn A → (B ∈ ∪ (𝐹 “ A) ↔ ∃x ∈ A B ∈ (𝐹‘x)))
Theorem	elunirn 5326*	Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
⊢ (Fun 𝐹 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))
Theorem	fnunirn 5327*	Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝐹 Fn 𝐼 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ 𝐼 A ∈ (𝐹‘x)))
Theorem	dff13 5328*	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y)))
Theorem	f1veqaeq 5329	If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	dff13f 5330*	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
⊢ Ⅎx𝐹    &   ⊢ Ⅎy𝐹    ⇒   ⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y)))
Theorem	f1mpt 5331*	Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐹 = (x ∈ A ↦ 𝐶)    &   ⊢ (x = y → 𝐶 = 𝐷)    ⇒   ⊢ (𝐹:A–1-1→B ↔ (∀x ∈ A 𝐶 ∈ B ∧ ∀x ∈ A ∀y ∈ A (𝐶 = 𝐷 → x = y)))
Theorem	f1fveq 5332	Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → ((𝐹‘𝐶) = (𝐹‘𝐷) ↔ 𝐶 = 𝐷))
Theorem	f1elima 5333	Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐹:A–1-1→B ∧ 𝑋 ∈ A ∧ 𝑌 ⊆ A) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))
Theorem	f1imass 5334	Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
Theorem	f1imaeq 5335	Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))
Theorem	f1imapss 5336	Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷))
Theorem	dff1o6 5337*	A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
⊢ (𝐹:A–1-1-onto→B ↔ (𝐹 Fn A ∧ ran 𝐹 = B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y)))
Theorem	f1ocnvfv1 5338	The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶)
Theorem	f1ocnvfv2 5339	The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ B) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶)
Theorem	f1ocnvfv 5340	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶))
Theorem	f1ocnvfvb 5341	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A ∧ 𝐷 ∈ B) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶))
Theorem	f1ocnvdm 5342	The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ B) → (◡𝐹‘𝐶) ∈ A)
Theorem	f1ocnvfvrneq 5343	If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	fcof1 5344	An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹:A⟶B ∧ (𝑅 ∘ 𝐹) = ( I ↾ A)) → 𝐹:A–1-1→B)
Theorem	fcofo 5345	An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ ((𝐹:A⟶B ∧ 𝑆:B⟶A ∧ (𝐹 ∘ 𝑆) = ( I ↾ B)) → 𝐹:A–onto→B)
Theorem	cbvfo 5346*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹‘x) = y → (φ ↔ ψ))    ⇒   ⊢ (𝐹:A–onto→B → (∀x ∈ A φ ↔ ∀y ∈ B ψ))
Theorem	cbvexfo 5347*	Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
⊢ ((𝐹‘x) = y → (φ ↔ ψ))    ⇒   ⊢ (𝐹:A–onto→B → (∃x ∈ A φ ↔ ∃y ∈ B ψ))
Theorem	cocan1 5348	An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹:B–1-1→𝐶 ∧ 𝐻:A⟶B ∧ 𝐾:A⟶B) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))
Theorem	cocan2 5349	A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ ((𝐹:A–onto→B ∧ 𝐻 Fn B ∧ 𝐾 Fn B) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))
Theorem	fcof1o 5350	Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (((𝐹:A⟶B ∧ 𝐺:B⟶A) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ B) ∧ (𝐺 ∘ 𝐹) = ( I ↾ A))) → (𝐹:A–1-1-onto→B ∧ ◡𝐹 = 𝐺))
Theorem	foeqcnvco 5351	Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ ((𝐹:A–onto→B ∧ 𝐺:A–onto→B) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ B)))
Theorem	f1eqcocnv 5352	Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ ((𝐹:A–1-1→B ∧ 𝐺:A–1-1→B) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ A)))
Theorem	fliftrel 5353*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆))
Theorem	fliftel 5354*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → (𝐶𝐹𝐷 ↔ ∃x ∈ 𝑋 (𝐶 = A ∧ 𝐷 = B)))
Theorem	fliftel1 5355*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ ((φ ∧ x ∈ 𝑋) → A𝐹B)
Theorem	fliftcnv 5356*	Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → ◡𝐹 = ran (x ∈ 𝑋 ↦ ⟨B, A⟩))
Theorem	fliftfun 5357*	The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    &   ⊢ (x = y → A = 𝐶)    &   ⊢ (x = y → B = 𝐷)    ⇒   ⊢ (φ → (Fun 𝐹 ↔ ∀x ∈ 𝑋 ∀y ∈ 𝑋 (A = 𝐶 → B = 𝐷)))
Theorem	fliftfund 5358*	The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    &   ⊢ (x = y → A = 𝐶)    &   ⊢ (x = y → B = 𝐷)    &   ⊢ ((φ ∧ (x ∈ 𝑋 ∧ y ∈ 𝑋 ∧ A = 𝐶)) → B = 𝐷)    ⇒   ⊢ (φ → Fun 𝐹)
Theorem	fliftfuns 5359*	The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → (Fun 𝐹 ↔ ∀y ∈ 𝑋 ∀z ∈ 𝑋 (⦋y / x⦌A = ⦋z / x⦌A → ⦋y / x⦌B = ⦋z / x⦌B)))
Theorem	fliftf 5360*	The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    ⇒   ⊢ (φ → (Fun 𝐹 ↔ 𝐹:ran (x ∈ 𝑋 ↦ A)⟶𝑆))
Theorem	fliftval 5361*	The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ ⟨A, B⟩)    &   ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅)    &   ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆)    &   ⊢ (x = 𝑌 → A = 𝐶)    &   ⊢ (x = 𝑌 → B = 𝐷)    &   ⊢ (φ → Fun 𝐹)    ⇒   ⊢ ((φ ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷)
Theorem	isoeq1 5362	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B)))
Theorem	isoeq2 5363	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑇, 𝑆 (A, B)))
Theorem	isoeq3 5364	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑇 (A, B)))
Theorem	isoeq4 5365	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (A = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, B)))
Theorem	isoeq5 5366	Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
⊢ (B = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (A, 𝐶)))
Theorem	nfiso 5367	Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ Ⅎx𝐻    &   ⊢ Ⅎx𝑅    &   ⊢ Ⅎx𝑆    &   ⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ Ⅎx 𝐻 Isom 𝑅, 𝑆 (A, B)
Theorem	isof1o 5368	An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A–1-1-onto→B)
Theorem	isorel 5369	An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	isoresbr 5370*	A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
⊢ ((𝐹 ↾ A) Isom 𝑅, 𝑆 (A, (𝐹 “ A)) → ∀x ∈ A ∀y ∈ A (x𝑅y → (𝐹‘x)𝑆(𝐹‘y)))
Theorem	isoid 5371	Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ ( I ↾ A) Isom 𝑅, 𝑅 (A, A)
Theorem	isocnv 5372	Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → ◡𝐻 Isom 𝑆, 𝑅 (B, A))
Theorem	isocnv2 5373	Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(A, B))
Theorem	isores2 5374	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (B × B))(A, B))
Theorem	isores1 5375	An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom (𝑅 ∩ (A × A)), 𝑆(A, B))
Theorem	isores3 5376	Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))
Theorem	isotr 5377	Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐺 Isom 𝑆, 𝑇 (B, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (A, 𝐶))
Theorem	isoini 5378	Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = (B ∩ (◡𝑆 “ {(𝐻‘𝐷)})))
Theorem	isoini2 5379	Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
⊢ 𝐶 = (A ∩ (◡𝑅 “ {𝑋}))    &   ⊢ 𝐷 = (B ∩ (◡𝑆 “ {(𝐻‘𝑋)}))    ⇒   ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝑋 ∈ A) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))
Theorem	isoselem 5380*	Lemma for isose 5381. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (φ → 𝐻 Isom 𝑅, 𝑆 (A, B))    &   ⊢ (φ → (𝐻 “ x) ∈ V)    ⇒   ⊢ (φ → (𝑅 Se A → 𝑆 Se B))
Theorem	isose 5381	An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑅 Se A ↔ 𝑆 Se B))
Theorem	isopolem 5382	Lemma for isopo 5383. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Po B → 𝑅 Po A))
Theorem	isopo 5383	An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑅 Po A ↔ 𝑆 Po B))
Theorem	isosolem 5384	Lemma for isoso 5385. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑆 Or B → 𝑅 Or A))
Theorem	isoso 5385	An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝑅 Or A ↔ 𝑆 Or B))
Theorem	f1oiso 5386*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
⊢ ((𝐻:A–1-1-onto→B ∧ 𝑆 = {⟨z, w⟩ ∣ ∃x ∈ A ∃y ∈ A ((z = (𝐻‘x) ∧ w = (𝐻‘y)) ∧ x𝑅y)}) → 𝐻 Isom 𝑅, 𝑆 (A, B))
Theorem	f1oiso2 5387*	Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝑆 = {⟨x, y⟩ ∣ ((x ∈ B ∧ y ∈ B) ∧ (◡𝐻‘x)𝑅(◡𝐻‘y))}    ⇒   ⊢ (𝐻:A–1-1-onto→B → 𝐻 Isom 𝑅, 𝑆 (A, B))
2.6.9  Restricted iota (description binder)
Syntax	crio 5388	Extend class notation with restricted description binder.
class (℩x ∈ A φ)
Definition	df-riota 5389	Define restricted description binder. In case there is no unique x such that (x ∈ A ∧ φ) holds, it evaluates to the empty set. See also comments for df-iota 4790. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
⊢ (℩x ∈ A φ) = (℩x(x ∈ A ∧ φ))
Theorem	riotaeqdv 5390*	Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → (℩x ∈ A ψ) = (℩x ∈ B ψ))
Theorem	riotabidv 5391*	Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (℩x ∈ A ψ) = (℩x ∈ A χ))
Theorem	riotaeqbidv 5392*	Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
⊢ (φ → A = B)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (℩x ∈ A ψ) = (℩x ∈ B χ))
Theorem	riotav 5393	An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
⊢ (℩x ∈ V φ) = (℩xφ)
Theorem	riotauni 5394	Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
⊢ (∃!x ∈ A φ → (℩x ∈ A φ) = ∪ {x ∈ A ∣ φ})
Theorem	nfriota1 5395*	The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎx(℩x ∈ A φ)
Theorem	nfriotadxy 5396*	Deduction version of nfriota 5397. (Contributed by Jim Kingdon, 12-Jan-2019.)
⊢ Ⅎyφ    &   ⊢ (φ → Ⅎxψ)    &   ⊢ (φ → ℲxA)    ⇒   ⊢ (φ → Ⅎx(℩y ∈ A ψ))
Theorem	nfriota 5397*	A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
⊢ Ⅎxφ    &   ⊢ ℲxA    ⇒   ⊢ Ⅎx(℩y ∈ A φ)
Theorem	cbvriota 5398*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎyφ    &   ⊢ Ⅎxψ    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (℩x ∈ A φ) = (℩y ∈ A ψ)
Theorem	cbvriotav 5399*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (℩x ∈ A φ) = (℩y ∈ A ψ)
Theorem	csbriotag 5400*	Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
⊢ (A ∈ 𝑉 → ⦋A / x⦌(℩y ∈ B φ) = (℩y ∈ B [A / x]φ))