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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmcnvcnv 4501 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4714). (Contributed by NM, 8-Apr-2007.)
dom A = dom A

Theoremrncnvcnv 4502 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
ran A = ran A

Theoremelreldm 4503 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 4504 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
(A = B → ran A = ran B)

Theoremrneqi 4505 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
A = B       ran A = ran B

Theoremrneqd 4506 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
(φA = B)       (φ → ran A = ran B)

Theoremrnss 4507 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
(AB → ran A ⊆ ran B)

Theorembrelrng 4508 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 4509 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 4510 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 4511 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 4512 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅 A𝑅B) → A dom 𝑅)

Theoremrelelrn 4513 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
((Rel 𝑅 A𝑅B) → B ran 𝑅)

Theoremreleldmb 4514* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (A dom 𝑅x A𝑅x))

Theoremrelelrnb 4515* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
(Rel 𝑅 → (A ran 𝑅x x𝑅A))

Theoremreleldmi 4516 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (A𝑅BA dom 𝑅)

Theoremrelelrni 4517 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
Rel 𝑅       (A𝑅BB ran 𝑅)

Theoremdfrnf 4518* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA    &   yA       ran A = {yx xAy}

Theoremelrn2 4519* Membership in a range. (Contributed by NM, 10-Jul-1994.)
A V       (A ran Bxx, A B)

Theoremelrn 4520* Membership in a range. (Contributed by NM, 2-Apr-2004.)
A V       (A ran Bx xBA)

Theoremnfdm 4521 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xdom A

Theoremnfrn 4522 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
xA       xran A

Theoremdmiin 4523 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
dom x A B x A dom B

Theoremrnopab 4524* 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 4525* 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 4526* 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 4527* 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 4528 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
𝐹 = (x AB)       ((x A B 𝑉) → B ran 𝐹)

Theoremelrnmptg 4529* 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 4530* 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 4531 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 4532 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 4533 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 4534 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 4535 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 4536* 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 4537 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
(Rel A → (A = ∅ ↔ ran A = ∅))

Theoremdmrnssfld 4538 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 4539 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 4540 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 4541 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 4542 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 4543 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 4544 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 4545 Range of a composition. (Contributed by NM, 19-Mar-1998.)
ran (AB) ⊆ ran A

Theoremdmcosseq 4546 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 4547 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
(dom A = ran B → dom (AB) = dom B)

Theoremrncoeq 4548 Range of a composition. (Contributed by NM, 19-Mar-1998.)
(dom A = ran B → ran (AB) = ran A)

Theoremreseq1 4549 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
(A = B → (A𝐶) = (B𝐶))

Theoremreseq2 4550 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
(A = B → (𝐶A) = (𝐶B))

Theoremreseq1i 4551 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
A = B       (A𝐶) = (B𝐶)

Theoremreseq2i 4552 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
A = B       (𝐶A) = (𝐶B)

Theoremreseq12i 4553 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)

Theoremreseq1d 4554 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(φA = B)       (φ → (A𝐶) = (B𝐶))

Theoremreseq2d 4555 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
(φA = B)       (φ → (𝐶A) = (𝐶B))

Theoremreseq12d 4556 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))

Theoremnfres 4557 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
xA    &   xB       x(AB)

Theoremcsbresg 4558 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(A 𝑉A / x(B𝐶) = (A / xBA / x𝐶))

Theoremres0 4559 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(A ↾ ∅) = ∅

Theoremopelres 4560 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 4561 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
B V       (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷))

Theoremopelresg 4562 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 4563 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(B 𝑉 → (A(𝐶𝐷)B ↔ (A𝐶B A 𝐷)))

Theoremopres 4564 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 4565 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 4566 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 4567 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((AB) ↾ 𝐶) = (A ↾ (B𝐶))

Theoremresundi 4568 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(A ↾ (B𝐶)) = ((AB) ∪ (A𝐶))

Theoremresundir 4569 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((AB) ↾ 𝐶) = ((A𝐶) ∪ (B𝐶))

Theoremresindi 4570 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(A ↾ (B𝐶)) = ((AB) ∩ (A𝐶))

Theoremresindir 4571 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((AB) ↾ 𝐶) = ((A𝐶) ∩ (B𝐶))

Theoreminres 4572 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(A ∩ (B𝐶)) = ((AB) ↾ 𝐶)

Theoremresiun1 4573* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
( x A B𝐶) = x A (B𝐶)

Theoremresiun2 4574* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 x A B) = x A (𝐶B)

Theoremdmres 4575 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (AB) = (B ∩ dom A)

Theoremssdmres 4576 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(A ⊆ dom B ↔ dom (BA) = A)

Theoremdmresexg 4577 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(B 𝑉 → dom (AB) V)

Theoremresss 4578 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(AB) ⊆ A

Theoremrescom 4579 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((AB) ↾ 𝐶) = ((A𝐶) ↾ B)

Theoremssres 4580 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(AB → (A𝐶) ⊆ (B𝐶))

Theoremssres2 4581 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(AB → (𝐶A) ⊆ (𝐶B))

Theoremrelres 4582 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 4583 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(B𝐶 → ((A𝐶) ↾ B) = (AB))

Theoremresabs2 4584 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(B𝐶 → ((AB) ↾ 𝐶) = (AB))

Theoremresidm 4585 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((AB) ↾ B) = (AB)

Theoremresima 4586 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((AB) “ B) = (AB)

Theoremresima2 4587 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
(B𝐶 → ((A𝐶) “ B) = (AB))

Theoremxpssres 4588 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶A → ((A × B) ↾ 𝐶) = (𝐶 × B))

Theoremelres 4589* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(A (B𝐶) ↔ x 𝐶 y(A = ⟨x, yx, y B))

Theoremelsnres 4590* 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 4591 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel A dom AB) → (AB) = A)

Theoremresdm 4592 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel A → (A ↾ dom A) = A)

Theoremresexg 4593 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 4594 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
A V       (AB) V

Theoremresopab 4595* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨x, y⟩ ∣ φ} ↾ A) = {⟨x, y⟩ ∣ (x A φ)}

Theoremresiexg 4596 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 4597 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 4598* 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 4599* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(BA → ((x A𝐶) ↾ B) = (x B𝐶))

Theoremresmpt3 4600* 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)

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