Theorem brcup 34302

Description: Binary relation form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brcup.1	⊢ 𝐴 ∈ V
brcup.2	⊢ 𝐵 ∈ V
brcup.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
brcup	⊢ (⟨𝐴, 𝐵⟩Cup𝐶 ↔ 𝐶 = (𝐴 ∪ 𝐵))

Proof of Theorem brcup

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

Step	Hyp	Ref	Expression
1		opex 5396	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcup.3	. 2 ⊢ 𝐶 ∈ V
3		df-cup 34232	. 2 ⊢ Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗ V)))
4		brcup.1	. . . 4 ⊢ 𝐴 ∈ V
5		brcup.2	. . . 4 ⊢ 𝐵 ∈ V
6	4, 5	opelvv 5644	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5652	. . 3 ⊢ (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
8	6, 2, 7	mpbir2an 708	. 2 ⊢ ⟨𝐴, 𝐵⟩((V × V) × V)𝐶
9		epel 5514	. . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		vex 3445	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10, 1	brcnv 5809	. . . . . . . 8 ⊢ (𝑦◡1st ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
12	4, 5	br1steq 33847	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩1st 𝑦 ↔ 𝑦 = 𝐴)
13	11, 12	bitri 274	. . . . . . 7 ⊢ (𝑦◡1st ⟨𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
14	9, 13	anbi12ci 628	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
15	14	exbii 1849	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
16		vex 3445	. . . . . 6 ⊢ 𝑥 ∈ V
17	16, 1	brco 5797	. . . . 5 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩))
18	4	clel3 3601	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
19	15, 17, 18	3bitr4i 302	. . . 4 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ 𝐴)
20	10, 1	brcnv 5809	. . . . . . . 8 ⊢ (𝑦◡2nd ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
21	4, 5	br2ndeq 33848	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩2nd 𝑦 ↔ 𝑦 = 𝐵)
22	20, 21	bitri 274	. . . . . . 7 ⊢ (𝑦◡2nd ⟨𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
23	9, 22	anbi12ci 628	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
24	23	exbii 1849	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
25	16, 1	brco 5797	. . . . 5 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩))
26	5	clel3 3601	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
27	24, 25, 26	3bitr4i 302	. . . 4 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ 𝐵)
28	19, 27	orbi12i 912	. . 3 ⊢ ((𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∨ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
29		brun 5136	. . 3 ⊢ (𝑥((◡1st ∘ E ) ∪ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∨ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩))
30		elun 4093	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
31	28, 29, 30	3bitr4ri 303	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥((◡1st ∘ E ) ∪ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩)
32	1, 2, 3, 8, 31	brtxpsd3 34259	1 ⊢ (⟨𝐴, 𝐵⟩Cup𝐶 ↔ 𝐶 = (𝐴 ∪ 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3441   ∪ cun 3894  ⟨cop 4575   class class class wbr 5085   E cep 5510   × cxp 5603  ^◡ccnv 5604   ∘ ccom 5609  1^st c1st 7872  2^nd c2nd 7873  Cupccup 34209

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-symdif 4186  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-eprel 5511  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fo 6469  df-fv 6471  df-1st 7874  df-2nd 7875  df-txp 34217  df-cup 34232

This theorem is referenced by:  brsuccf  34304