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Theorem brapply 31722
Description: The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
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 4874 . . . 4 {(𝐴 “ {𝐵})} ∈ V
21inex1 4764 . . 3 ({(𝐴 “ {𝐵})} ∩ Singletons ) ∈ V
3 unieq 4415 . . . . 5 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
43unieqd 4417 . . . 4 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
54eqeq2d 2631 . . 3 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → (𝐶 = 𝑥𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
62, 5ceqsexv 3231 . 2 (∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥) ↔ 𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7 df-apply 31656 . . . 4 Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
87breqi 4624 . . 3 (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ⟨𝐴, 𝐵⟩(( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶)
9 opex 4898 . . . 4 𝐴, 𝐵⟩ ∈ V
10 brapply.3 . . . 4 𝐶 ∈ V
119, 10brco 5257 . . 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 3192 . . . . . . 7 𝑥 ∈ V
139, 12brco 5257 . . . . . 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 3192 . . . . . . . . . 10 𝑦 ∈ V
159, 14brco 5257 . . . . . . . . 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 3192 . . . . . . . . . . . . 13 𝑧 ∈ V
1916, 17, 18brpprod3a 31670 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ ∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏))
20 3anrot 1041 . . . . . . . . . . . . . 14 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ (𝐴 I 𝑎𝐵Singleton𝑏𝑧 = ⟨𝑎, 𝑏⟩))
21 vex 3192 . . . . . . . . . . . . . . . . 17 𝑎 ∈ V
2221ideq 5239 . . . . . . . . . . . . . . . 16 (𝐴 I 𝑎𝐴 = 𝑎)
23 eqcom 2628 . . . . . . . . . . . . . . . 16 (𝐴 = 𝑎𝑎 = 𝐴)
2422, 23bitri 264 . . . . . . . . . . . . . . 15 (𝐴 I 𝑎𝑎 = 𝐴)
25 vex 3192 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
2617, 25brsingle 31701 . . . . . . . . . . . . . . 15 (𝐵Singleton𝑏𝑏 = {𝐵})
27 biid 251 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
2824, 26, 273anbi123i 1249 . . . . . . . . . . . . . 14 ((𝐴 I 𝑎𝐵Singleton𝑏𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
2920, 28bitri 264 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
30292exbii 1772 . . . . . . . . . . . 12 (∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
31 snex 4874 . . . . . . . . . . . . 13 {𝐵} ∈ V
32 opeq1 4375 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3332eqeq2d 2631 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, 𝑏⟩))
34 opeq2 4376 . . . . . . . . . . . . . 14 (𝑏 = {𝐵} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐵}⟩)
3534eqeq2d 2631 . . . . . . . . . . . . 13 (𝑏 = {𝐵} → (𝑧 = ⟨𝐴, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, {𝐵}⟩))
3616, 31, 33, 35ceqsex2v 3234 . . . . . . . . . . . 12 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
3719, 30, 363bitri 286 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧 = ⟨𝐴, {𝐵}⟩)
3837anbi1i 730 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦) ↔ (𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
3938exbii 1771 . . . . . . . . 9 (∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦) ↔ ∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
40 opex 4898 . . . . . . . . . . 11 𝐴, {𝐵}⟩ ∈ V
41 breq1 4621 . . . . . . . . . . 11 (𝑧 = ⟨𝐴, {𝐵}⟩ → (𝑧(Singleton ∘ Img)𝑦 ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦))
4240, 41ceqsexv 3231 . . . . . . . . . 10 (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦)
4340, 14brco 5257 . . . . . . . . . 10 (⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦 ↔ ∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦))
4416, 31, 12brimg 31721 . . . . . . . . . . . . 13 (⟨𝐴, {𝐵}⟩Img𝑥𝑥 = (𝐴 “ {𝐵}))
4512, 14brsingle 31701 . . . . . . . . . . . . 13 (𝑥Singleton𝑦𝑦 = {𝑥})
4644, 45anbi12i 732 . . . . . . . . . . . 12 ((⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ (𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
4746exbii 1771 . . . . . . . . . . 11 (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ ∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
48 imaexg 7057 . . . . . . . . . . . . 13 (𝐴 ∈ V → (𝐴 “ {𝐵}) ∈ V)
4916, 48ax-mp 5 . . . . . . . . . . . 12 (𝐴 “ {𝐵}) ∈ V
50 sneq 4163 . . . . . . . . . . . . 13 (𝑥 = (𝐴 “ {𝐵}) → {𝑥} = {(𝐴 “ {𝐵})})
5150eqeq2d 2631 . . . . . . . . . . . 12 (𝑥 = (𝐴 “ {𝐵}) → (𝑦 = {𝑥} ↔ 𝑦 = {(𝐴 “ {𝐵})}))
5249, 51ceqsexv 3231 . . . . . . . . . . 11 (∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5347, 52bitri 264 . . . . . . . . . 10 (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5442, 43, 533bitri 286 . . . . . . . . 9 (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5515, 39, 543bitri 286 . . . . . . . 8 (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦 = {(𝐴 “ {𝐵})})
56 eqid 2621 . . . . . . . . 9 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))
57 brxp 5112 . . . . . . . . . 10 (𝑦(V × V)𝑥 ↔ (𝑦 ∈ V ∧ 𝑥 ∈ V))
5814, 12, 57mpbir2an 954 . . . . . . . . 9 𝑦(V × V)𝑥
59 epel 4993 . . . . . . . . . . 11 (𝑧 E 𝑦𝑧𝑦)
6059anbi1i 730 . . . . . . . . . 10 ((𝑧 E 𝑦𝑧 Singletons ) ↔ (𝑧𝑦𝑧 Singletons ))
6114brres 5367 . . . . . . . . . 10 (𝑧( E ↾ Singletons )𝑦 ↔ (𝑧 E 𝑦𝑧 Singletons ))
62 elin 3779 . . . . . . . . . 10 (𝑧 ∈ (𝑦 Singletons ) ↔ (𝑧𝑦𝑧 Singletons ))
6360, 61, 623bitr4ri 293 . . . . . . . . 9 (𝑧 ∈ (𝑦 Singletons ) ↔ 𝑧( E ↾ Singletons )𝑦)
6414, 12, 56, 58, 63brtxpsd3 31680 . . . . . . . 8 (𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥𝑥 = (𝑦 Singletons ))
6555, 64anbi12i 732 . . . . . . 7 ((⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))𝑥) ↔ (𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 Singletons )))
6665exbii 1771 . . . . . 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 )))
67 ineq1 3790 . . . . . . . 8 (𝑦 = {(𝐴 “ {𝐵})} → (𝑦 Singletons ) = ({(𝐴 “ {𝐵})} ∩ Singletons ))
6867eqeq2d 2631 . . . . . . 7 (𝑦 = {(𝐴 “ {𝐵})} → (𝑥 = (𝑦 Singletons ) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
691, 68ceqsexv 3231 . . . . . 6 (∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 Singletons )) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7013, 66, 693bitri 286 . . . . 5 (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7112, 10brco 5257 . . . . . 6 (𝑥( Bigcup Bigcup )𝐶 ↔ ∃𝑦(𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶))
7214brbigcup 31682 . . . . . . . . 9 (𝑥 Bigcup 𝑦 𝑥 = 𝑦)
73 eqcom 2628 . . . . . . . . 9 ( 𝑥 = 𝑦𝑦 = 𝑥)
7472, 73bitri 264 . . . . . . . 8 (𝑥 Bigcup 𝑦𝑦 = 𝑥)
7510brbigcup 31682 . . . . . . . . 9 (𝑦 Bigcup 𝐶 𝑦 = 𝐶)
76 eqcom 2628 . . . . . . . . 9 ( 𝑦 = 𝐶𝐶 = 𝑦)
7775, 76bitri 264 . . . . . . . 8 (𝑦 Bigcup 𝐶𝐶 = 𝑦)
7874, 77anbi12i 732 . . . . . . 7 ((𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶) ↔ (𝑦 = 𝑥𝐶 = 𝑦))
7978exbii 1771 . . . . . 6 (∃𝑦(𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶) ↔ ∃𝑦(𝑦 = 𝑥𝐶 = 𝑦))
80 vuniex 6914 . . . . . . 7 𝑥 ∈ V
81 unieq 4415 . . . . . . . 8 (𝑦 = 𝑥 𝑦 = 𝑥)
8281eqeq2d 2631 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = 𝑦𝐶 = 𝑥))
8380, 82ceqsexv 3231 . . . . . 6 (∃𝑦(𝑦 = 𝑥𝐶 = 𝑦) ↔ 𝐶 = 𝑥)
8471, 79, 833bitri 286 . . . . 5 (𝑥( Bigcup Bigcup )𝐶𝐶 = 𝑥)
8570, 84anbi12i 732 . . . 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))
8685exbii 1771 . . 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))
878, 11, 863bitri 286 . 2 (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥))
88 dffv5 31708 . . 3 (𝐴𝐵) = ({(𝐴 “ {𝐵})} ∩ Singletons )
8988eqeq2i 2633 . 2 (𝐶 = (𝐴𝐵) ↔ 𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
906, 87, 893bitr4i 292 1 (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cdif 3556  cin 3558  csymdif 3826  {csn 4153  cop 4159   cuni 4407   class class class wbr 4618   E cep 4988   I cid 4989   × cxp 5077  ran crn 5080  cres 5081  cima 5082  ccom 5083  cfv 5852  ctxp 31613  pprodcpprod 31614   Bigcup cbigcup 31617  Singletoncsingle 31621   Singletons csingles 31622  Imgcimg 31625  Applycapply 31628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-pprod 31638  df-bigcup 31641  df-singleton 31645  df-singles 31646  df-image 31647  df-cart 31648  df-img 31649  df-apply 31656
This theorem is referenced by:  dfrecs2  31734  dfrdg4  31735
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