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Theorem brcap 31681
Description: Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcap.1 𝐴 ∈ V
brcap.2 𝐵 ∈ V
brcap.3 𝐶 ∈ V
Assertion
Ref Expression
brcap (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))

Proof of Theorem brcap
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4898 . 2 𝐴, 𝐵⟩ ∈ V
2 brcap.3 . 2 𝐶 ∈ V
3 df-cap 31610 . 2 Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))
4 brcap.1 . . . 4 𝐴 ∈ V
5 brcap.2 . . . 4 𝐵 ∈ V
64, 5opelvv 5131 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5112 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 954 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 epel 4993 . . . . . . 7 (𝑥 E 𝑦𝑥𝑦)
10 vex 3194 . . . . . . . . 9 𝑦 ∈ V
1110, 1brcnv 5270 . . . . . . . 8 (𝑦1st𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
124, 5, 10br1steq 31366 . . . . . . . 8 (⟨𝐴, 𝐵⟩1st 𝑦𝑦 = 𝐴)
1311, 12bitri 264 . . . . . . 7 (𝑦1st𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
149, 13anbi12ci 733 . . . . . 6 ((𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴𝑥𝑦))
1514exbii 1772 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
16 vex 3194 . . . . . 6 𝑥 ∈ V
1716, 1brco 5257 . . . . 5 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩))
184clel3 3329 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
1915, 17, 183bitr4i 292 . . . 4 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐴)
2010, 1brcnv 5270 . . . . . . . 8 (𝑦2nd𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
214, 5, 10br2ndeq 31367 . . . . . . . 8 (⟨𝐴, 𝐵⟩2nd 𝑦𝑦 = 𝐵)
2220, 21bitri 264 . . . . . . 7 (𝑦2nd𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
239, 22anbi12ci 733 . . . . . 6 ((𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵𝑥𝑦))
2423exbii 1772 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2516, 1brco 5257 . . . . 5 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩))
265clel3 3329 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2724, 25, 263bitr4i 292 . . . 4 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐵)
2819, 27anbi12i 732 . . 3 ((𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥𝐴𝑥𝐵))
29 brin 4669 . . 3 (𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩))
30 elin 3779 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4ri 293 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩)
321, 2, 3, 8, 31brtxpsd3 31637 1 (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  Vcvv 3191  cin 3559  cop 4159   class class class wbr 4618   E cep 4988   × cxp 5077  ccnv 5078  ccom 5083  1st c1st 7114  2nd c2nd 7115  Capccap 31587
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3827  df-nul 3897  df-if 4064  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-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fo 5856  df-fv 5858  df-1st 7116  df-2nd 7117  df-txp 31594  df-cap 31610
This theorem is referenced by:  brrestrict  31690
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