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Theorem brcap 35569
Description: Binary relation 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 5470 . 2 𝐴, 𝐵⟩ ∈ V
2 brcap.3 . 2 𝐶 ∈ V
3 df-cap 35499 . 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 5722 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5731 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 709 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 epel 5589 . . . . . . 7 (𝑥 E 𝑦𝑥𝑦)
10 vex 3477 . . . . . . . . 9 𝑦 ∈ V
1110, 1brcnv 5889 . . . . . . . 8 (𝑦1st𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
124, 5br1steq 35399 . . . . . . . 8 (⟨𝐴, 𝐵⟩1st 𝑦𝑦 = 𝐴)
1311, 12bitri 274 . . . . . . 7 (𝑦1st𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
149, 13anbi12ci 627 . . . . . 6 ((𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴𝑥𝑦))
1514exbii 1842 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
16 vex 3477 . . . . . 6 𝑥 ∈ V
1716, 1brco 5877 . . . . 5 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩))
184clel3 3651 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
1915, 17, 183bitr4i 302 . . . 4 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐴)
2010, 1brcnv 5889 . . . . . . . 8 (𝑦2nd𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
214, 5br2ndeq 35400 . . . . . . . 8 (⟨𝐴, 𝐵⟩2nd 𝑦𝑦 = 𝐵)
2220, 21bitri 274 . . . . . . 7 (𝑦2nd𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
239, 22anbi12ci 627 . . . . . 6 ((𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵𝑥𝑦))
2423exbii 1842 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2516, 1brco 5877 . . . . 5 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩))
265clel3 3651 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2724, 25, 263bitr4i 302 . . . 4 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐵)
2819, 27anbi12i 626 . . 3 ((𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥𝐴𝑥𝐵))
29 brin 5204 . . 3 (𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩))
30 elin 3965 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4ri 303 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩)
321, 2, 3, 8, 31brtxpsd3 35525 1 (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  cin 3948  cop 4638   class class class wbr 5152   E cep 5585   × cxp 5680  ccnv 5681  ccom 5686  1st c1st 7997  2nd c2nd 7998  Capccap 35476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4245  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-1st 7999  df-2nd 8000  df-txp 35483  df-cap 35499
This theorem is referenced by:  brrestrict  35578
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