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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfdm 4501 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xdom A
 
Theoremnfrn 4502 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xran A
 
Theoremdmiin 4503 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom x A B x A dom B
 
Theoremrnopab 4504* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
ran {⟨x, y⟩ ∣ φ} = {yxφ}
 
Theoremrnmpt 4505* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (x AB)       ran 𝐹 = {yx A y = B}
 
Theoremelrnmpt 4506* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
𝐹 = (x AB)       (𝐶 𝑉 → (𝐶 ran 𝐹x A 𝐶 = B))
 
Theoremelrnmpt1s 4507* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝐹 = (x AB)    &   (x = 𝐷B = 𝐶)       ((𝐷 A 𝐶 𝑉) → 𝐶 ran 𝐹)
 
Theoremelrnmpt1 4508 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (x AB)       ((x A B 𝑉) → B ran 𝐹)
 
Theoremelrnmptg 4509* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (x AB)       (x A B 𝑉 → (𝐶 ran 𝐹x A 𝐶 = B))
 
Theoremelrnmpti 4510* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐹 = (x AB)    &   B V       (𝐶 ran 𝐹x A 𝐶 = B)
 
Theoremrn0 4511 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)
ran ∅ = ∅
 
Theoremdfiun3g 4512 Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(x A B 𝐶 x A B = ran (x AB))
 
Theoremdfiin3g 4513 Alternate definition of indexed intersection when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
(x A B 𝐶 x A B = ran (x AB))
 
Theoremdfiun3 4514 Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
B V        x A B = ran (x AB)
 
Theoremdfiin3 4515 Alternate definition of indexed intersection when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
B V        x A B = ran (x AB)
 
Theoremriinint 4516* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑋 𝑉 𝑘 𝐼 𝑆𝑋) → (𝑋 𝑘 𝐼 𝑆) = ({𝑋} ∪ ran (𝑘 𝐼𝑆)))
 
Theoremrelrn0 4517 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel A → (A = ∅ ↔ ran A = ∅))
 
Theoremdmrnssfld 4518 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
(dom A ∪ ran A) ⊆ A
 
Theoremdmexg 4519 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
(A 𝑉 → dom A V)
 
Theoremrnexg 4520 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
(A 𝑉 → ran A V)
 
Theoremdmex 4521 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
A V       dom A V
 
Theoremrnex 4522 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
A V       ran A V
 
Theoremiprc 4523 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set. (Contributed by NM, 1-Jan-2007.)
¬ I V
 
Theoremdmcoss 4524 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
dom (AB) ⊆ dom B
 
Theoremrncoss 4525 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (AB) ⊆ ran A
 
Theoremdmcosseq 4526 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran B ⊆ dom A → dom (AB) = dom B)
 
Theoremdmcoeq 4527 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom A = ran B → dom (AB) = dom B)
 
Theoremrncoeq 4528 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom A = ran B → ran (AB) = ran A)
 
Theoremreseq1 4529 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(A = B → (A𝐶) = (B𝐶))
 
Theoremreseq2 4530 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(A = B → (𝐶A) = (𝐶B))
 
Theoremreseq1i 4531 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
A = B       (A𝐶) = (B𝐶)
 
Theoremreseq2i 4532 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = B       (𝐶A) = (𝐶B)
 
Theoremreseq12i 4533 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)
 
Theoremreseq1d 4534 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(φA = B)       (φ → (A𝐶) = (B𝐶))
 
Theoremreseq2d 4535 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (𝐶A) = (𝐶B))
 
Theoremreseq12d 4536 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))
 
Theoremnfres 4537 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
xA    &   xB       x(AB)
 
Theoremcsbresg 4538 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(A 𝑉A / x(B𝐶) = (A / xBA / x𝐶))
 
Theoremres0 4539 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(A ↾ ∅) = ∅
 
Theoremopelres 4540 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
B V       (⟨A, B (𝐶𝐷) ↔ (⟨A, B 𝐶 A 𝐷))
 
Theorembrres 4541 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
B V       (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷))
 
Theoremopelresg 4542 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
(B 𝑉 → (⟨A, B (𝐶𝐷) ↔ (⟨A, B 𝐶 A 𝐷)))
 
Theorembrresg 4543 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(B 𝑉 → (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷)))
 
Theoremopres 4544 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
B V       (A 𝐷 → (⟨A, B (𝐶𝐷) ↔ ⟨A, B 𝐶))
 
Theoremresieq 4545 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
((B A 𝐶 A) → (B( I ↾ A)𝐶B = 𝐶))
 
Theoremopelresi 4546 A, A belongs to a restriction of the identity class iff A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
(A 𝑉 → (⟨A, A ( I ↾ B) ↔ A B))
 
Theoremresres 4547 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((AB) ↾ 𝐶) = (A ↾ (B𝐶))
 
Theoremresundi 4548 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(A ↾ (B𝐶)) = ((AB) ∪ (A𝐶))
 
Theoremresundir 4549 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((AB) ↾ 𝐶) = ((A𝐶) ∪ (B𝐶))
 
Theoremresindi 4550 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(A ↾ (B𝐶)) = ((AB) ∩ (A𝐶))
 
Theoremresindir 4551 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((AB) ↾ 𝐶) = ((A𝐶) ∩ (B𝐶))
 
Theoreminres 4552 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(A ∩ (B𝐶)) = ((AB) ↾ 𝐶)
 
Theoremresiun1 4553* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
( x A B𝐶) = x A (B𝐶)
 
Theoremresiun2 4554* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 x A B) = x A (𝐶B)
 
Theoremdmres 4555 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (AB) = (B ∩ dom A)
 
Theoremssdmres 4556 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(A ⊆ dom B ↔ dom (BA) = A)
 
Theoremdmresexg 4557 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(B 𝑉 → dom (AB) V)
 
Theoremresss 4558 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(AB) ⊆ A
 
Theoremrescom 4559 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((AB) ↾ 𝐶) = ((A𝐶) ↾ B)
 
Theoremssres 4560 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(AB → (A𝐶) ⊆ (B𝐶))
 
Theoremssres2 4561 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB → (𝐶A) ⊆ (𝐶B))
 
Theoremrelres 4562 A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Rel (AB)
 
Theoremresabs1 4563 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(B𝐶 → ((A𝐶) ↾ B) = (AB))
 
Theoremresabs2 4564 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(B𝐶 → ((AB) ↾ 𝐶) = (AB))
 
Theoremresidm 4565 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((AB) ↾ B) = (AB)
 
Theoremresima 4566 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((AB) “ B) = (AB)
 
Theoremresima2 4567 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
(B𝐶 → ((A𝐶) “ B) = (AB))
 
Theoremxpssres 4568 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶A → ((A × B) ↾ 𝐶) = (𝐶 × B))
 
Theoremelres 4569* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(A (B𝐶) ↔ x 𝐶 y(A = ⟨x, yx, y B))
 
Theoremelsnres 4570* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
𝐶 V       (A (B ↾ {𝐶}) ↔ y(A = ⟨𝐶, y𝐶, y B))
 
Theoremrelssres 4571 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel A dom AB) → (AB) = A)
 
Theoremresdm 4572 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel A → (A ↾ dom A) = A)
 
Theoremresexg 4573 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A 𝑉 → (AB) V)
 
Theoremresex 4574 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
A V       (AB) V
 
Theoremresopab 4575* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨x, y⟩ ∣ φ} ↾ A) = {⟨x, y⟩ ∣ (x A φ)}
 
Theoremresiexg 4576 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
(A 𝑉 → ( I ↾ A) V)
 
Theoremiss 4577 A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(A ⊆ I ↔ A = ( I ↾ dom A))
 
Theoremresopab2 4578* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
(AB → ({⟨x, y⟩ ∣ (x B φ)} ↾ A) = {⟨x, y⟩ ∣ (x A φ)})
 
Theoremresmpt 4579* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(BA → ((x A𝐶) ↾ B) = (x B𝐶))
 
Theoremresmpt3 4580* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
((x A𝐶) ↾ B) = (x (AB) ↦ 𝐶)
 
Theoremdfres2 4581* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅A) = {⟨x, y⟩ ∣ (x A x𝑅y)}
 
Theoremopabresid 4582* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨x, y⟩ ∣ (x A y = x)} = ( I ↾ A)
 
Theoremmptresid 4583* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
(x Ax) = ( I ↾ A)
 
Theoremdmresi 4584 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ A) = A
 
Theoremresid 4585 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel A → (A ↾ V) = A)
 
Theoremimaeq1 4586 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(A = B → (A𝐶) = (B𝐶))
 
Theoremimaeq2 4587 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(A = B → (𝐶A) = (𝐶B))
 
Theoremimaeq1i 4588 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
A = B       (A𝐶) = (B𝐶)
 
Theoremimaeq2i 4589 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
A = B       (𝐶A) = (𝐶B)
 
Theoremimaeq1d 4590 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(φA = B)       (φ → (A𝐶) = (B𝐶))
 
Theoremimaeq2d 4591 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(φA = B)       (φ → (𝐶A) = (𝐶B))
 
Theoremimaeq12d 4592 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))
 
Theoremdfima2 4593* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = {yx B xAy}
 
Theoremdfima3 4594* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB) = {yx(x B x, y A)}
 
Theoremelimag 4595* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(A 𝑉 → (A (B𝐶) ↔ x 𝐶 xBA))
 
Theoremelima 4596* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
A V       (A (B𝐶) ↔ x 𝐶 xBA)
 
Theoremelima2 4597* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
A V       (A (B𝐶) ↔ x(x 𝐶 xBA))
 
Theoremelima3 4598* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
A V       (A (B𝐶) ↔ x(x 𝐶 x, A B))
 
Theoremnfima 4599 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
xA    &   xB       x(AB)
 
Theoremnfimad 4600 Deduction version of bound-variable hypothesis builder nfima 4599. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φxA)    &   (φxB)       (φx(AB))
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