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Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxpriindim 4401* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(y y A → (𝐶 × (𝐷 x A B)) = ((𝐶 × 𝐷) ∩ x A (𝐶 × B)))

Theoremeliunxp 4402* Membership in a union of cross products. Analogue of elxp 4289 for nonconstant B(x). (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐶 x A ({x} × B) ↔ xy(𝐶 = ⟨x, y (x A y B)))

Theoremopeliunxp2 4403* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)
(x = 𝐶B = 𝐸)       (⟨𝐶, 𝐷 x A ({x} × B) ↔ (𝐶 A 𝐷 𝐸))

Theoremraliunxp 4404* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4406, B(y) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
(x = ⟨y, z⟩ → (φψ))       (x y A ({y} × B)φy A z B ψ)

Theoremrexiunxp 4405* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4407, B(y) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
(x = ⟨y, z⟩ → (φψ))       (x y A ({y} × B)φy A z B ψ)

Theoremralxp 4406* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
(x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremrexxp 4407* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
(x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremdjussxp 4408* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
x A ({x} × B) ⊆ (A × V)

Theoremralxpf 4409* Version of ralxp 4406 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
yφ    &   zφ    &   xψ    &   (x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremrexxpf 4410* Version of rexxp 4407 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
yφ    &   zφ    &   xψ    &   (x = ⟨y, z⟩ → (φψ))       (x (A × B)φy A z B ψ)

Theoremiunxpf 4411* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
y𝐶    &   z𝐶    &   x𝐷    &   (x = ⟨y, z⟩ → 𝐶 = 𝐷)        x (A × B)𝐶 = y A z B 𝐷

Theoremopabbi2dv 4412* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2138. (Contributed by NM, 24-Feb-2014.)
Rel A    &   (φ → (⟨x, y Aψ))       (φA = {⟨x, y⟩ ∣ ψ})

Theoremrelop 4413* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
A V    &   B V       (Rel ⟨A, B⟩ ↔ xy(A = {x} B = {x, y}))

Theoremideqg 4414 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(B 𝑉 → (A I BA = B))

Theoremideq 4415 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
B V       (A I BA = B)

Theoremididg 4416 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A 𝑉A I A)

Theoremissetid 4417 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
(A V ↔ A I A)

Theoremcoss1 4418 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
(AB → (A𝐶) ⊆ (B𝐶))

Theoremcoss2 4419 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
(AB → (𝐶A) ⊆ (𝐶B))

Theoremcoeq1 4420 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(A = B → (A𝐶) = (B𝐶))

Theoremcoeq2 4421 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
(A = B → (𝐶A) = (𝐶B))

Theoremcoeq1i 4422 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
A = B       (A𝐶) = (B𝐶)

Theoremcoeq2i 4423 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
A = B       (𝐶A) = (𝐶B)

Theoremcoeq1d 4424 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(φA = B)       (φ → (A𝐶) = (B𝐶))

Theoremcoeq2d 4425 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
(φA = B)       (φ → (𝐶A) = (𝐶B))

Theoremcoeq12i 4426 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)

Theoremcoeq12d 4427 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))

Theoremnfco 4428 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
xA    &   xB       x(AB)

Theorembrcog 4429* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))

Theoremopelco2g 4430* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
((A 𝑉 B 𝑊) → (⟨A, B (𝐶𝐷) ↔ x(⟨A, x 𝐷 x, B 𝐶)))

Theorembrcogw 4431 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(((A 𝑉 B 𝑊 𝑋 𝑍) (A𝐷𝑋 𝑋𝐶B)) → A(𝐶𝐷)B)

Theoremeqbrrdva 4432* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(φA ⊆ (𝐶 × 𝐷))    &   (φB ⊆ (𝐶 × 𝐷))    &   ((φ x 𝐶 y 𝐷) → (xAyxBy))       (φA = B)

Theorembrco 4433* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
A V    &   B V       (A(𝐶𝐷)Bx(A𝐷x x𝐶B))

Theoremopelco 4434* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
A V    &   B V       (⟨A, B (𝐶𝐷) ↔ x(A𝐷x x𝐶B))

Theoremcnvss 4435 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
(ABAB)

Theoremcnveq 4436 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
(A = BA = B)

Theoremcnveqi 4437 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
A = B       A = B

Theoremcnveqd 4438 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
(φA = B)       (φA = B)

Theoremelcnv 4439* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
(A 𝑅xy(A = ⟨x, y y𝑅x))

Theoremelcnv2 4440* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
(A 𝑅xy(A = ⟨x, yy, x 𝑅))

Theoremnfcnv 4441 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xA

Theoremopelcnvg 4442 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((A 𝐶 B 𝐷) → (⟨A, B 𝑅 ↔ ⟨B, A 𝑅))

Theorembrcnvg 4443 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))

Theoremopelcnv 4444 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
A V    &   B V       (⟨A, B 𝑅 ↔ ⟨B, A 𝑅)

Theorembrcnv 4445 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
A V    &   B V       (A𝑅BB𝑅A)

Theoremcsbcnvg 4446 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
(A 𝑉A / x𝐹 = A / x𝐹)

Theoremcnvco 4447 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = (BA)

Theoremcnvuni 4448* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
A = x A x

Theoremdfdm3 4449* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
dom A = {xyx, y A}

Theoremdfrn2 4450* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
ran A = {yx xAy}

Theoremdfrn3 4451* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
ran A = {yxx, y A}

Theoremelrn2g 4452* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(A 𝑉 → (A ran Bxx, A B))

Theoremelrng 4453* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
(A 𝑉 → (A ran Bx xBA))

Theoremdfdm4 4454 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
dom A = ran A

Theoremdfdmf 4455* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       dom A = {xy xAy}

Theoremcsbdmg 4456 Distribute proper substitution through the domain of a class. (Contributed by Jim Kingdon, 8-Dec-2018.)
(A 𝑉A / xdom B = dom A / xB)

Theoremeldmg 4457* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A dom By ABy))

Theoremeldm2g 4458* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A dom ByA, y B))

Theoremeldm 4459* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
A V       (A dom By ABy)

Theoremeldm2 4460* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
A V       (A dom ByA, y B)

Theoremdmss 4461 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
(AB → dom A ⊆ dom B)

Theoremdmeq 4462 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
(A = B → dom A = dom B)

Theoremdmeqi 4463 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
A = B       dom A = dom B

Theoremdmeqd 4464 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
(φA = B)       (φ → dom A = dom B)

Theoremopeldm 4465 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
A V    &   B V       (⟨A, B 𝐶A dom 𝐶)

Theorembreldm 4466 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
A V    &   B V       (A𝑅BA dom 𝑅)

Theoremopeldmg 4467 Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.)
((A 𝑉 B 𝑊) → (⟨A, B 𝐶A dom 𝐶))

Theorembreldmg 4468 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
((A 𝐶 B 𝐷 A𝑅B) → A dom 𝑅)

Theoremdmun 4469 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom (AB) = (dom A ∪ dom B)

Theoremdmin 4470 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
dom (AB) ⊆ (dom A ∩ dom B)

Theoremdmiun 4471 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
dom x A B = x A dom B

Theoremdmuni 4472* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
dom A = x A dom x

Theoremdmopab 4473* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
dom {⟨x, y⟩ ∣ φ} = {xyφ}

Theoremdmopabss 4474* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
dom {⟨x, y⟩ ∣ (x A φ)} ⊆ A

Theoremdmopab3 4475* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
(x A yφ ↔ dom {⟨x, y⟩ ∣ (x A φ)} = A)

Theoremdm0 4476 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom ∅ = ∅

Theoremdmi 4477 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom I = V

Theoremdmv 4478 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
dom V = V

Theoremdm0rn0 4479 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
(dom A = ∅ ↔ ran A = ∅)

Theoremreldm0 4480 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
(Rel A → (A = ∅ ↔ dom A = ∅))

Theoremdmmrnm 4481* A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
(x x dom Ay y ran A)

Theoremdmxpm 4482* The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(x x B → dom (A × B) = A)

Theoremdmxpinm 4483* The domain of the intersection of two square cross products. Unlike dmin 4470, equality holds. (Contributed by NM, 29-Jan-2008.)
(x x (AB) → dom ((A × A) ∩ (B × B)) = (AB))

Theoremxpid11m 4484* The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)
((x x A x x B) → ((A × A) = (B × B) ↔ A = B))

Theoremdmcnvcnv 4485 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4698). (Contributed by NM, 8-Apr-2007.)
dom A = dom A

Theoremrncnvcnv 4486 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
ran A = ran A

Theoremelreldm 4487 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
((Rel A B A) → B dom A)

Theoremrneq 4488 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(A = B → ran A = ran B)

Theoremrneqi 4489 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
A = B       ran A = ran B

Theoremrneqd 4490 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(φA = B)       (φ → ran A = ran B)

Theoremrnss 4491 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(AB → ran A ⊆ ran B)

Theorembrelrng 4492 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
((A 𝐹 B 𝐺 A𝐶B) → B ran 𝐶)

Theoremopelrng 4493 Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
((A 𝐹 B 𝐺 A, B 𝐶) → B ran 𝐶)

Theorembrelrn 4494 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
A V    &   B V       (A𝐶BB ran 𝐶)

Theoremopelrn 4495 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
A V    &   B V       (⟨A, B 𝐶B ran 𝐶)

Theoremreleldm 4496 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅 A𝑅B) → A dom 𝑅)

Theoremrelelrn 4497 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅 A𝑅B) → B ran 𝑅)

Theoremreleldmb 4498* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (A dom 𝑅x A𝑅x))

Theoremrelelrnb 4499* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (A ran 𝑅x x𝑅A))

Theoremreleldmi 4500 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (A𝑅BA dom 𝑅)

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