Theorem brcap 35958

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 5439	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcap.3	. 2 ⊢ 𝐶 ∈ V
3		df-cap 35888	. 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 5694	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5703	. . 3 ⊢ (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
8	6, 2, 7	mpbir2an 711	. 2 ⊢ ⟨𝐴, 𝐵⟩((V × V) × V)𝐶
9		epel 5556	. . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		vex 3463	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10, 1	brcnv 5862	. . . . . . . 8 ⊢ (𝑦◡1st ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
12	4, 5	br1steq 35788	. . . . . . . 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 1848	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
16		vex 3463	. . . . . 6 ⊢ 𝑥 ∈ V
17	16, 1	brco 5850	. . . . 5 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩))
18	4	clel3 3641	. . . . 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 5862	. . . . . . . 8 ⊢ (𝑦◡2nd ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
21	4, 5	br2ndeq 35789	. . . . . . . 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 1848	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
25	16, 1	brco 5850	. . . . 5 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩))
26	5	clel3 3641	. . . . 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 5171	. . 3 ⊢ (𝑥((◡1st ∘ E ) ∩ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∧ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩))
30		elin 3942	. . 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 35914	1 ⊢ (⟨𝐴, 𝐵⟩Cap𝐶 ↔ 𝐶 = (𝐴 ∩ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459   ∩ cin 3925  ⟨cop 4607   class class class wbr 5119   E cep 5552   × cxp 5652  ^◡ccnv 5653   ∘ ccom 5658  1^st c1st 7986  2^nd c2nd 7987  Capccap 35865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-symdif 4228  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-eprel 5553  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-1st 7988  df-2nd 7989  df-txp 35872  df-cap 35888

This theorem is referenced by:  brrestrict  35967