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Theorem brapply 35920
Description: Binary relation form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Hypotheses
Ref Expression
brapply.1 𝐴 ∈ V
brapply.2 𝐵 ∈ V
brapply.3 𝐶 ∈ V
Assertion
Ref Expression
brapply (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))

Proof of Theorem brapply
Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5442 . . . 4 {(𝐴 “ {𝐵})} ∈ V
21inex1 5323 . . 3 ({(𝐴 “ {𝐵})} ∩ Singletons ) ∈ V
3 unieq 4923 . . . . 5 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
43unieqd 4925 . . . 4 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
54eqeq2d 2746 . . 3 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → (𝐶 = 𝑥𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
62, 5ceqsexv 3530 . 2 (∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥) ↔ 𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7 df-apply 35855 . . . 4 Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
87breqi 5154 . . 3 (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ⟨𝐴, 𝐵⟩(( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶)
9 opex 5475 . . . 4 𝐴, 𝐵⟩ ∈ V
10 brapply.3 . . . 4 𝐶 ∈ V
119, 10brco 5884 . . 3 (⟨𝐴, 𝐵⟩(( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶 ↔ ∃𝑥(⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥𝑥( Bigcup Bigcup )𝐶))
12 vex 3482 . . . . . . 7 𝑥 ∈ V
139, 12brco 5884 . . . . . 6 (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥 ↔ ∃𝑦(⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥))
14 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
159, 14brco 5884 . . . . . . . . 9 (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦 ↔ ∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦))
16 brapply.1 . . . . . . . . . . . . 13 𝐴 ∈ V
17 brapply.2 . . . . . . . . . . . . 13 𝐵 ∈ V
18 vex 3482 . . . . . . . . . . . . 13 𝑧 ∈ V
1916, 17, 18brpprod3a 35868 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ ∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏))
20 3anrot 1099 . . . . . . . . . . . . . 14 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ (𝐴 I 𝑎𝐵Singleton𝑏𝑧 = ⟨𝑎, 𝑏⟩))
21 vex 3482 . . . . . . . . . . . . . . . . 17 𝑎 ∈ V
2221ideq 5866 . . . . . . . . . . . . . . . 16 (𝐴 I 𝑎𝐴 = 𝑎)
23 eqcom 2742 . . . . . . . . . . . . . . . 16 (𝐴 = 𝑎𝑎 = 𝐴)
2422, 23bitri 275 . . . . . . . . . . . . . . 15 (𝐴 I 𝑎𝑎 = 𝐴)
25 vex 3482 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
2617, 25brsingle 35899 . . . . . . . . . . . . . . 15 (𝐵Singleton𝑏𝑏 = {𝐵})
27 biid 261 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
2824, 26, 273anbi123i 1154 . . . . . . . . . . . . . 14 ((𝐴 I 𝑎𝐵Singleton𝑏𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
2920, 28bitri 275 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
30292exbii 1846 . . . . . . . . . . . 12 (∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
31 snex 5442 . . . . . . . . . . . . 13 {𝐵} ∈ V
32 opeq1 4878 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3332eqeq2d 2746 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, 𝑏⟩))
34 opeq2 4879 . . . . . . . . . . . . . 14 (𝑏 = {𝐵} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐵}⟩)
3534eqeq2d 2746 . . . . . . . . . . . . 13 (𝑏 = {𝐵} → (𝑧 = ⟨𝐴, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, {𝐵}⟩))
3616, 31, 33, 35ceqsex2v 3536 . . . . . . . . . . . 12 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
3719, 30, 363bitri 297 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧 = ⟨𝐴, {𝐵}⟩)
3837anbi1i 624 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦) ↔ (𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
3938exbii 1845 . . . . . . . . 9 (∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦) ↔ ∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
40 opex 5475 . . . . . . . . . . 11 𝐴, {𝐵}⟩ ∈ V
41 breq1 5151 . . . . . . . . . . 11 (𝑧 = ⟨𝐴, {𝐵}⟩ → (𝑧(Singleton ∘ Img)𝑦 ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦))
4240, 41ceqsexv 3530 . . . . . . . . . 10 (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦)
4340, 14brco 5884 . . . . . . . . . 10 (⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦 ↔ ∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦))
4416, 31, 12brimg 35919 . . . . . . . . . . . . 13 (⟨𝐴, {𝐵}⟩Img𝑥𝑥 = (𝐴 “ {𝐵}))
4512, 14brsingle 35899 . . . . . . . . . . . . 13 (𝑥Singleton𝑦𝑦 = {𝑥})
4644, 45anbi12i 628 . . . . . . . . . . . 12 ((⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ (𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
4746exbii 1845 . . . . . . . . . . 11 (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ ∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
4816imaex 7937 . . . . . . . . . . . 12 (𝐴 “ {𝐵}) ∈ V
49 sneq 4641 . . . . . . . . . . . . 13 (𝑥 = (𝐴 “ {𝐵}) → {𝑥} = {(𝐴 “ {𝐵})})
5049eqeq2d 2746 . . . . . . . . . . . 12 (𝑥 = (𝐴 “ {𝐵}) → (𝑦 = {𝑥} ↔ 𝑦 = {(𝐴 “ {𝐵})}))
5148, 50ceqsexv 3530 . . . . . . . . . . 11 (∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5247, 51bitri 275 . . . . . . . . . 10 (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5342, 43, 523bitri 297 . . . . . . . . 9 (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5415, 39, 533bitri 297 . . . . . . . 8 (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦 = {(𝐴 “ {𝐵})})
55 eqid 2735 . . . . . . . . 9 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))
56 brxp 5738 . . . . . . . . . 10 (𝑦(V × V)𝑥 ↔ (𝑦 ∈ V ∧ 𝑥 ∈ V))
5714, 12, 56mpbir2an 711 . . . . . . . . 9 𝑦(V × V)𝑥
58 epel 5592 . . . . . . . . . . 11 (𝑧 E 𝑦𝑧𝑦)
5958anbi1ci 626 . . . . . . . . . 10 ((𝑧 Singletons 𝑧 E 𝑦) ↔ (𝑧𝑦𝑧 Singletons ))
6014brresi 6009 . . . . . . . . . 10 (𝑧( E ↾ Singletons )𝑦 ↔ (𝑧 Singletons 𝑧 E 𝑦))
61 elin 3979 . . . . . . . . . 10 (𝑧 ∈ (𝑦 Singletons ) ↔ (𝑧𝑦𝑧 Singletons ))
6259, 60, 613bitr4ri 304 . . . . . . . . 9 (𝑧 ∈ (𝑦 Singletons ) ↔ 𝑧( E ↾ Singletons )𝑦)
6314, 12, 55, 57, 62brtxpsd3 35878 . . . . . . . 8 (𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥𝑥 = (𝑦 Singletons ))
6454, 63anbi12i 628 . . . . . . 7 ((⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥) ↔ (𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 Singletons )))
6564exbii 1845 . . . . . 6 (∃𝑦(⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥) ↔ ∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 Singletons )))
66 ineq1 4221 . . . . . . . 8 (𝑦 = {(𝐴 “ {𝐵})} → (𝑦 Singletons ) = ({(𝐴 “ {𝐵})} ∩ Singletons ))
6766eqeq2d 2746 . . . . . . 7 (𝑦 = {(𝐴 “ {𝐵})} → (𝑥 = (𝑦 Singletons ) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
681, 67ceqsexv 3530 . . . . . 6 (∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 Singletons )) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
6913, 65, 683bitri 297 . . . . 5 (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7012, 10brco 5884 . . . . . 6 (𝑥( Bigcup Bigcup )𝐶 ↔ ∃𝑦(𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶))
7114brbigcup 35880 . . . . . . . . 9 (𝑥 Bigcup 𝑦 𝑥 = 𝑦)
72 eqcom 2742 . . . . . . . . 9 ( 𝑥 = 𝑦𝑦 = 𝑥)
7371, 72bitri 275 . . . . . . . 8 (𝑥 Bigcup 𝑦𝑦 = 𝑥)
7410brbigcup 35880 . . . . . . . . 9 (𝑦 Bigcup 𝐶 𝑦 = 𝐶)
75 eqcom 2742 . . . . . . . . 9 ( 𝑦 = 𝐶𝐶 = 𝑦)
7674, 75bitri 275 . . . . . . . 8 (𝑦 Bigcup 𝐶𝐶 = 𝑦)
7773, 76anbi12i 628 . . . . . . 7 ((𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶) ↔ (𝑦 = 𝑥𝐶 = 𝑦))
7877exbii 1845 . . . . . 6 (∃𝑦(𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶) ↔ ∃𝑦(𝑦 = 𝑥𝐶 = 𝑦))
79 vuniex 7758 . . . . . . 7 𝑥 ∈ V
80 unieq 4923 . . . . . . . 8 (𝑦 = 𝑥 𝑦 = 𝑥)
8180eqeq2d 2746 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = 𝑦𝐶 = 𝑥))
8279, 81ceqsexv 3530 . . . . . 6 (∃𝑦(𝑦 = 𝑥𝐶 = 𝑦) ↔ 𝐶 = 𝑥)
8370, 78, 823bitri 297 . . . . 5 (𝑥( Bigcup Bigcup )𝐶𝐶 = 𝑥)
8469, 83anbi12i 628 . . . 4 ((⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥𝑥( Bigcup Bigcup )𝐶) ↔ (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥))
8584exbii 1845 . . 3 (∃𝑥(⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥𝑥( Bigcup Bigcup )𝐶) ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥))
868, 11, 853bitri 297 . 2 (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥))
87 dffv5 35906 . . 3 (𝐴𝐵) = ({(𝐴 “ {𝐵})} ∩ Singletons )
8887eqeq2i 2748 . 2 (𝐶 = (𝐴𝐵) ↔ 𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
896, 86, 883bitr4i 303 1 (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cdif 3960  cin 3962  csymdif 4258  {csn 4631  cop 4637   cuni 4912   class class class wbr 5148   I cid 5582   E cep 5588   × cxp 5687  ran crn 5690  cres 5691  cima 5692  ccom 5693  cfv 6563  ctxp 35812  pprodcpprod 35813   Bigcup cbigcup 35816  Singletoncsingle 35820   Singletons csingles 35821  Imgcimg 35824  Applycapply 35827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-pprod 35837  df-bigcup 35840  df-singleton 35844  df-singles 35845  df-image 35846  df-cart 35847  df-img 35848  df-apply 35855
This theorem is referenced by:  dfrecs2  35932  dfrdg4  35933
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