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Theorem brcap 33409
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 5332 . 2 𝐴, 𝐵⟩ ∈ V
2 brcap.3 . 2 𝐶 ∈ V
3 df-cap 33339 . 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 5570 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5577 . . 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 5445 . . . . . . 7 (𝑥 E 𝑦𝑥𝑦)
10 vex 3476 . . . . . . . . 9 𝑦 ∈ V
1110, 1brcnv 5729 . . . . . . . 8 (𝑦1st𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
124, 5br1steq 33022 . . . . . . . 8 (⟨𝐴, 𝐵⟩1st 𝑦𝑦 = 𝐴)
1311, 12bitri 277 . . . . . . 7 (𝑦1st𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
149, 13anbi12ci 629 . . . . . 6 ((𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴𝑥𝑦))
1514exbii 1848 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
16 vex 3476 . . . . . 6 𝑥 ∈ V
1716, 1brco 5717 . . . . 5 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩))
184clel3 3634 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
1915, 17, 183bitr4i 305 . . . 4 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐴)
2010, 1brcnv 5729 . . . . . . . 8 (𝑦2nd𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
214, 5br2ndeq 33023 . . . . . . . 8 (⟨𝐴, 𝐵⟩2nd 𝑦𝑦 = 𝐵)
2220, 21bitri 277 . . . . . . 7 (𝑦2nd𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
239, 22anbi12ci 629 . . . . . 6 ((𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵𝑥𝑦))
2423exbii 1848 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2516, 1brco 5717 . . . . 5 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩))
265clel3 3634 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2724, 25, 263bitr4i 305 . . . 4 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐵)
2819, 27anbi12i 628 . . 3 ((𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥𝐴𝑥𝐵))
29 brin 5094 . . 3 (𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩))
30 elin 3929 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4ri 306 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥((1st ∘ E ) ∩ (2nd ∘ E ))⟨𝐴, 𝐵⟩)
321, 2, 3, 8, 31brtxpsd3 33365 1 (⟨𝐴, 𝐵⟩Cap𝐶𝐶 = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  wex 1780  wcel 2114  Vcvv 3473  cin 3912  cop 4549   class class class wbr 5042   E cep 5440   × cxp 5529  ccnv 5530  ccom 5535  1st c1st 7665  2nd c2nd 7666  Capccap 33316
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-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-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-cap 33339
This theorem is referenced by:  brrestrict  33418
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