Theorem brcap 34912

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⟩Cap𝐶 ↔ 𝐶 = (𝐴 ∩ 𝐵))

Proof of Theorem brcap

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5465	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcap.3	. 2 ⊢ 𝐶 ∈ V
3		df-cap 34842	. 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
6	4, 5	opelvv 5717	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5726	. . 3 ⊢ (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
8	6, 2, 7	mpbir2an 710	. 2 ⊢ ⟨𝐴, 𝐵⟩((V × V) × V)𝐶
9		epel 5584	. . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		vex 3479	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10, 1	brcnv 5883	. . . . . . . 8 ⊢ (𝑦◡1st ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
12	4, 5	br1steq 34742	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩1st 𝑦 ↔ 𝑦 = 𝐴)
13	11, 12	bitri 275	. . . . . . 7 ⊢ (𝑦◡1st ⟨𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
14	9, 13	anbi12ci 629	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
15	14	exbii 1851	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
16		vex 3479	. . . . . 6 ⊢ 𝑥 ∈ V
17	16, 1	brco 5871	. . . . 5 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩))
18	4	clel3 3652	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
19	15, 17, 18	3bitr4i 303	. . . 4 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ 𝐴)
20	10, 1	brcnv 5883	. . . . . . . 8 ⊢ (𝑦◡2nd ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
21	4, 5	br2ndeq 34743	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩2nd 𝑦 ↔ 𝑦 = 𝐵)
22	20, 21	bitri 275	. . . . . . 7 ⊢ (𝑦◡2nd ⟨𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
23	9, 22	anbi12ci 629	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
24	23	exbii 1851	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
25	16, 1	brco 5871	. . . . 5 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩))
26	5	clel3 3652	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
27	24, 25, 26	3bitr4i 303	. . . 4 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ 𝐵)
28	19, 27	anbi12i 628	. . 3 ⊢ ((𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
29		brin 5201	. . 3 ⊢ (𝑥((◡1st ∘ E ) ∩ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩))
30		elin 3965	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
31	28, 29, 30	3bitr4ri 304	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥((◡1st ∘ E ) ∩ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩)
32	1, 2, 3, 8, 31	brtxpsd3 34868	1 ⊢ (⟨𝐴, 𝐵⟩Cap𝐶 ↔ 𝐶 = (𝐴 ∩ 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948  ⟨cop 4635   class class class wbr 5149   E cep 5580   × cxp 5675  ^◡ccnv 5676   ∘ ccom 5681  1^st c1st 7973  2^nd c2nd 7974  Capccap 34819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-cap 34842

This theorem is referenced by:  brrestrict  34921