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Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnvcnv3 5501* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

Theoremdfrel2 5502 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
(Rel 𝑅𝑅 = 𝑅)

Theoremdfrel4v 5503* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6151 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
(Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Theoremdfrel4 5504* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6151 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
𝑥𝑅    &   𝑦𝑅       (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Theoremcnvcnv 5505 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)
𝐴 = (𝐴 ∩ (V × V))

Theoremcnvcnv2 5506 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
𝐴 = (𝐴 ↾ V)

Theoremcnvcnvss 5507 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
𝐴𝐴

Theoremcnveqb 5508 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))

Theoremcnveq0 5509 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
(Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))

Theoremdfrel3 5510 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
(Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Theoremdmresv 5511 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
dom (𝐴 ↾ V) = dom 𝐴

Theoremrnresv 5512 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
ran (𝐴 ↾ V) = ran 𝐴

Theoremdfrn4 5513 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
ran 𝐴 = (𝐴 “ V)

Theoremcsbrn 5514 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵

Theoremrescnvcnv 5515 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcnvcnvres 5516 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremimacnvcnv 5517 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremdmsnn0 5518 The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)

Theoremrnsnn0 5519 The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)
(𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

Theoremdmsn0 5520 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
dom {∅} = ∅

Theoremcnvsn0 5521 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
{∅} = ∅

Theoremdmsn0el 5522 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
(∅ ∈ 𝐴 → dom {𝐴} = ∅)

Theoremrelsn2 5523 A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)
𝐴 ∈ V       (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)

Theoremdmsnopg 5524 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

Theoremdmsnopss 5525 The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on 𝐵). (Contributed by Mario Carneiro, 30-Apr-2015.)
dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

Theoremdmpropg 5526 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})

Theoremdmsnop 5527 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V       dom {⟨𝐴, 𝐵⟩} = {𝐴}

Theoremdmprop 5528 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
𝐵 ∈ V    &   𝐷 ∈ V       dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Theoremdmtpop 5529 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
𝐵 ∈ V    &   𝐷 ∈ V    &   𝐹 ∈ V       dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

Theoremcnvcnvsn 5530 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5536, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Theoremdmsnsnsn 5531 The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
dom {{{𝐴}}} = {𝐴}

Theoremrnsnopg 5532 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Theoremrnpropg 5533 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Theoremrnsnop 5534 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ran {⟨𝐴, 𝐵⟩} = {𝐵}

Theoremop1sta 5535 Extract the first member of an ordered pair. (See op2nda 5538 to extract the second member, op1stb 4867 for an alternate version, and op1st 7067 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        dom {⟨𝐴, 𝐵⟩} = 𝐴

Theoremcnvsn 5536 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Theoremop2ndb 5537 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4867 to extract the first member, op2nda 5538 for an alternate version, and op2nd 7068 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        {⟨𝐴, 𝐵⟩} = 𝐵

Theoremop2nda 5538 Extract the second member of an ordered pair. (See op1sta 5535 to extract the first member, op2ndb 5537 for an alternate version, and op2nd 7068 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V        ran {⟨𝐴, 𝐵⟩} = 𝐵

Theoremcnvsng 5539 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Theoremopswap 5540 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)
{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴

Theoremcnvresima 5541 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)

Theoremresdm2 5542 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom 𝐴) = 𝐴

Theoremresdmres 5543 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)

Theoremimadmres 5544 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)

Theoremmptpreima 5545* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐹𝐶) = {𝑥𝐴𝐵𝐶}

Theoremmptiniseg 5546* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})

Theoremdmmpt 5547 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = {𝑥𝐴𝐵 ∈ V}

Theoremdmmptss 5548* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴

Theoremdmmptg 5549* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
(∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)

Theoremrelco 5550 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Rel (𝐴𝐵)

Theoremdfco2 5551* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
(𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))

Theoremdfco2a 5552* Generalization of dfco2 5551, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Theoremcoundi 5553 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremcoundir 5554 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremcores 5555 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))

Theoremresco 5556 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))

Theoremimaco 5557 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))

Theoremrnco 5558 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)

Theoremrnco2 5559 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
ran (𝐴𝐵) = (𝐴 “ ran 𝐵)

Theoremdmco 5560 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (𝐴𝐵) = (𝐵 “ dom 𝐴)

Theoremcoeq0 5561 A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 5553 and coundir 5554 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

Theoremcoiun 5562* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremcocnvcnv1 5563 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcocnvcnv2 5564 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcores2 5565 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
(dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))

Theoremco02 5566 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅

Theoremco01 5567 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅

Theoremcoi1 5568 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Theoremcoi2 5569 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Theoremcoires1 5570 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
(𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)

Theoremcoass 5571 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))

Theoremrelcnvtr 5572 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))

Theoremrelssdmrn 5573 A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Theoremcnvssrndm 5574 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Theoremcossxp 5575 Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Theoremrelrelss 5576 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Theoremunielrel 5577 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)

Theoremrelfld 5578 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))

Theoremrelresfld 5579 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)

Theoremrelcoi2 5580 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)

Theoremrelcoi1 5581 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.)
(Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)

Theoremunidmrn 5582 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)

Theoremrelcnvfld 5583 if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 𝑅 = 𝑅)

Theoremdfdm2 5584 Alternate definition of domain df-dm 5048 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)

Theoremunixp 5585 The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.)
((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))

Theoremunixp0 5586 A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)

Theoremunixpid 5587 Field of a square Cartesian product. (Contributed by FL, 10-Oct-2009.)
(𝐴 × 𝐴) = 𝐴

Theoremressn 5588 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Theoremcnviin 5589* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)

Theoremcnvpo 5590 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Po 𝐴𝑅 Po 𝐴)

Theoremcnvso 5591 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Or 𝐴𝑅 Or 𝐴)

Theoremxpco 5592 Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
(𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Theoremxpcoid 5593 Composition of two square Cartesian products. (Contributed by Thierry Arnoux, 14-Jan-2018.)
((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)

Theoremelsnxp 5594* Elementhood to a cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)
(𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

TheoremelsnxpOLD 5595* Obsolete proof of elsnxp 5594 as of 14-Jul-2021. (Contributed by Thierry Arnoux, 10-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

2.3.10  The Predecessor Class

Syntaxcpred 5596 The predecessors symbol.
class Pred(𝑅, 𝐴, 𝑋)

Definitiondf-pred 5597 Define the predecessor class of a relationship. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 5610) . (Contributed by Scott Fenton, 29-Jan-2011.)
Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))

Theorempredeq123 5598 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)
((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))

Theorempredeq1 5599 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))

Theorempredeq2 5600 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

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