Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brapply Structured version   Visualization version   GIF version

Theorem brapply 33407
 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 5308 . . . 4 {(𝐴 “ {𝐵})} ∈ V
21inex1 5197 . . 3 ({(𝐴 “ {𝐵})} ∩ Singletons ) ∈ V
3 unieq 4825 . . . . 5 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
43unieqd 4828 . . . 4 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
54eqeq2d 2831 . . 3 (𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) → (𝐶 = 𝑥𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
62, 5ceqsexv 3520 . 2 (∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥) ↔ 𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7 df-apply 33342 . . . 4 Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
87breqi 5048 . . 3 (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ⟨𝐴, 𝐵⟩(( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))𝐶)
9 opex 5332 . . . 4 𝐴, 𝐵⟩ ∈ V
10 brapply.3 . . . 4 𝐶 ∈ V
119, 10brco 5717 . . 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 3476 . . . . . . 7 𝑥 ∈ V
139, 12brco 5717 . . . . . 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 3476 . . . . . . . . . 10 𝑦 ∈ V
159, 14brco 5717 . . . . . . . . 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 3476 . . . . . . . . . . . . 13 𝑧 ∈ V
1916, 17, 18brpprod3a 33355 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧 ↔ ∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏))
20 3anrot 1096 . . . . . . . . . . . . . 14 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ (𝐴 I 𝑎𝐵Singleton𝑏𝑧 = ⟨𝑎, 𝑏⟩))
21 vex 3476 . . . . . . . . . . . . . . . . 17 𝑎 ∈ V
2221ideq 5699 . . . . . . . . . . . . . . . 16 (𝐴 I 𝑎𝐴 = 𝑎)
23 eqcom 2827 . . . . . . . . . . . . . . . 16 (𝐴 = 𝑎𝑎 = 𝐴)
2422, 23bitri 277 . . . . . . . . . . . . . . 15 (𝐴 I 𝑎𝑎 = 𝐴)
25 vex 3476 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
2617, 25brsingle 33386 . . . . . . . . . . . . . . 15 (𝐵Singleton𝑏𝑏 = {𝐵})
27 biid 263 . . . . . . . . . . . . . . 15 (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
2824, 26, 273anbi123i 1151 . . . . . . . . . . . . . 14 ((𝐴 I 𝑎𝐵Singleton𝑏𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
2920, 28bitri 277 . . . . . . . . . . . . 13 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ (𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
30292exbii 1849 . . . . . . . . . . . 12 (∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝐴 I 𝑎𝐵Singleton𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
31 snex 5308 . . . . . . . . . . . . 13 {𝐵} ∈ V
32 opeq1 4779 . . . . . . . . . . . . . 14 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
3332eqeq2d 2831 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, 𝑏⟩))
34 opeq2 4780 . . . . . . . . . . . . . 14 (𝑏 = {𝐵} → ⟨𝐴, 𝑏⟩ = ⟨𝐴, {𝐵}⟩)
3534eqeq2d 2831 . . . . . . . . . . . . 13 (𝑏 = {𝐵} → (𝑧 = ⟨𝐴, 𝑏⟩ ↔ 𝑧 = ⟨𝐴, {𝐵}⟩))
3616, 31, 33, 35ceqsex2v 3523 . . . . . . . . . . . 12 (∃𝑎𝑏(𝑎 = 𝐴𝑏 = {𝐵} ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝐴, {𝐵}⟩)
3719, 30, 363bitri 299 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧 = ⟨𝐴, {𝐵}⟩)
3837anbi1i 625 . . . . . . . . . 10 ((⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦) ↔ (𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
3938exbii 1848 . . . . . . . . 9 (∃𝑧(⟨𝐴, 𝐵⟩pprod( I , Singleton)𝑧𝑧(Singleton ∘ Img)𝑦) ↔ ∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦))
40 opex 5332 . . . . . . . . . . 11 𝐴, {𝐵}⟩ ∈ V
41 breq1 5045 . . . . . . . . . . 11 (𝑧 = ⟨𝐴, {𝐵}⟩ → (𝑧(Singleton ∘ Img)𝑦 ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦))
4240, 41ceqsexv 3520 . . . . . . . . . 10 (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ ⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦)
4340, 14brco 5717 . . . . . . . . . 10 (⟨𝐴, {𝐵}⟩(Singleton ∘ Img)𝑦 ↔ ∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦))
4416, 31, 12brimg 33406 . . . . . . . . . . . . 13 (⟨𝐴, {𝐵}⟩Img𝑥𝑥 = (𝐴 “ {𝐵}))
4512, 14brsingle 33386 . . . . . . . . . . . . 13 (𝑥Singleton𝑦𝑦 = {𝑥})
4644, 45anbi12i 628 . . . . . . . . . . . 12 ((⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ (𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
4746exbii 1848 . . . . . . . . . . 11 (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ ∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}))
4816imaex 7599 . . . . . . . . . . . 12 (𝐴 “ {𝐵}) ∈ V
49 sneq 4553 . . . . . . . . . . . . 13 (𝑥 = (𝐴 “ {𝐵}) → {𝑥} = {(𝐴 “ {𝐵})})
5049eqeq2d 2831 . . . . . . . . . . . 12 (𝑥 = (𝐴 “ {𝐵}) → (𝑦 = {𝑥} ↔ 𝑦 = {(𝐴 “ {𝐵})}))
5148, 50ceqsexv 3520 . . . . . . . . . . 11 (∃𝑥(𝑥 = (𝐴 “ {𝐵}) ∧ 𝑦 = {𝑥}) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5247, 51bitri 277 . . . . . . . . . 10 (∃𝑥(⟨𝐴, {𝐵}⟩Img𝑥𝑥Singleton𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5342, 43, 523bitri 299 . . . . . . . . 9 (∃𝑧(𝑧 = ⟨𝐴, {𝐵}⟩ ∧ 𝑧(Singleton ∘ Img)𝑦) ↔ 𝑦 = {(𝐴 “ {𝐵})})
5415, 39, 533bitri 299 . . . . . . . 8 (⟨𝐴, 𝐵⟩((Singleton ∘ Img) ∘ pprod( I , Singleton))𝑦𝑦 = {(𝐴 “ {𝐵})})
55 eqid 2820 . . . . . . . . 9 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))
56 brxp 5577 . . . . . . . . . 10 (𝑦(V × V)𝑥 ↔ (𝑦 ∈ V ∧ 𝑥 ∈ V))
5714, 12, 56mpbir2an 709 . . . . . . . . 9 𝑦(V × V)𝑥
58 epel 5445 . . . . . . . . . . 11 (𝑧 E 𝑦𝑧𝑦)
5958anbi1ci 627 . . . . . . . . . 10 ((𝑧 Singletons 𝑧 E 𝑦) ↔ (𝑧𝑦𝑧 Singletons ))
6014brresi 5838 . . . . . . . . . 10 (𝑧( E ↾ Singletons )𝑦 ↔ (𝑧 Singletons 𝑧 E 𝑦))
61 elin 3929 . . . . . . . . . 10 (𝑧 ∈ (𝑦 Singletons ) ↔ (𝑧𝑦𝑧 Singletons ))
6259, 60, 613bitr4ri 306 . . . . . . . . 9 (𝑧 ∈ (𝑦 Singletons ) ↔ 𝑧( E ↾ Singletons )𝑦)
6314, 12, 55, 57, 62brtxpsd3 33365 . . . . . . . 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 1848 . . . . . 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 4159 . . . . . . . 8 (𝑦 = {(𝐴 “ {𝐵})} → (𝑦 Singletons ) = ({(𝐴 “ {𝐵})} ∩ Singletons ))
6766eqeq2d 2831 . . . . . . 7 (𝑦 = {(𝐴 “ {𝐵})} → (𝑥 = (𝑦 Singletons ) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons )))
681, 67ceqsexv 3520 . . . . . 6 (∃𝑦(𝑦 = {(𝐴 “ {𝐵})} ∧ 𝑥 = (𝑦 Singletons )) ↔ 𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
6913, 65, 683bitri 299 . . . . 5 (⟨𝐴, 𝐵⟩(((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))𝑥𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
7012, 10brco 5717 . . . . . 6 (𝑥( Bigcup Bigcup )𝐶 ↔ ∃𝑦(𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶))
7114brbigcup 33367 . . . . . . . . 9 (𝑥 Bigcup 𝑦 𝑥 = 𝑦)
72 eqcom 2827 . . . . . . . . 9 ( 𝑥 = 𝑦𝑦 = 𝑥)
7371, 72bitri 277 . . . . . . . 8 (𝑥 Bigcup 𝑦𝑦 = 𝑥)
7410brbigcup 33367 . . . . . . . . 9 (𝑦 Bigcup 𝐶 𝑦 = 𝐶)
75 eqcom 2827 . . . . . . . . 9 ( 𝑦 = 𝐶𝐶 = 𝑦)
7674, 75bitri 277 . . . . . . . 8 (𝑦 Bigcup 𝐶𝐶 = 𝑦)
7773, 76anbi12i 628 . . . . . . 7 ((𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶) ↔ (𝑦 = 𝑥𝐶 = 𝑦))
7877exbii 1848 . . . . . 6 (∃𝑦(𝑥 Bigcup 𝑦𝑦 Bigcup 𝐶) ↔ ∃𝑦(𝑦 = 𝑥𝐶 = 𝑦))
79 vuniex 7443 . . . . . . 7 𝑥 ∈ V
80 unieq 4825 . . . . . . . 8 (𝑦 = 𝑥 𝑦 = 𝑥)
8180eqeq2d 2831 . . . . . . 7 (𝑦 = 𝑥 → (𝐶 = 𝑦𝐶 = 𝑥))
8279, 81ceqsexv 3520 . . . . . 6 (∃𝑦(𝑦 = 𝑥𝐶 = 𝑦) ↔ 𝐶 = 𝑥)
8370, 78, 823bitri 299 . . . . 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 1848 . . 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 299 . 2 (⟨𝐴, 𝐵⟩Apply𝐶 ↔ ∃𝑥(𝑥 = ({(𝐴 “ {𝐵})} ∩ Singletons ) ∧ 𝐶 = 𝑥))
87 dffv5 33393 . . 3 (𝐴𝐵) = ({(𝐴 “ {𝐵})} ∩ Singletons )
8887eqeq2i 2833 . 2 (𝐶 = (𝐴𝐵) ↔ 𝐶 = ({(𝐴 “ {𝐵})} ∩ Singletons ))
896, 86, 883bitr4i 305 1 (⟨𝐴, 𝐵⟩Apply𝐶𝐶 = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3473   ∖ cdif 3910   ∩ cin 3912   △ csymdif 4196  {csn 4543  ⟨cop 4549  ∪ cuni 4814   class class class wbr 5042   I cid 5435   E cep 5440   × cxp 5529  ran crn 5532   ↾ cres 5533   “ cima 5534   ∘ ccom 5535  ‘cfv 6331   ⊗ ctxp 33299  pprodcpprod 33300   Bigcup cbigcup 33303  Singletoncsingle 33307   Singletons csingles 33308  Imgcimg 33311  Applycapply 33314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-symdif 4197  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-eprel 5441  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fo 6337  df-fv 6339  df-1st 7667  df-2nd 7668  df-txp 33323  df-pprod 33324  df-bigcup 33327  df-singleton 33331  df-singles 33332  df-image 33333  df-cart 33334  df-img 33335  df-apply 33342 This theorem is referenced by:  dfrecs2  33419  dfrdg4  33420
 Copyright terms: Public domain W3C validator