Theorem brcup 33400

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 5355	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcup.3	. 2 ⊢ 𝐶 ∈ V
3		df-cup 33330	. 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 5593	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5600	. . 3 ⊢ (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
8	6, 2, 7	mpbir2an 709	. 2 ⊢ ⟨𝐴, 𝐵⟩((V × V) × V)𝐶
9		epel 5468	. . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		vex 3497	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10, 1	brcnv 5752	. . . . . . . 8 ⊢ (𝑦◡1st ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
12	4, 5	br1steq 33014	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩1st 𝑦 ↔ 𝑦 = 𝐴)
13	11, 12	bitri 277	. . . . . . 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 1844	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
16		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
17	16, 1	brco 5740	. . . . 5 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡1st ⟨𝐴, 𝐵⟩))
18	4	clel3 3654	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑥 ∈ 𝑦))
19	15, 17, 18	3bitr4i 305	. . . 4 ⊢ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ 𝐴)
20	10, 1	brcnv 5752	. . . . . . . 8 ⊢ (𝑦◡2nd ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
21	4, 5	br2ndeq 33015	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩2nd 𝑦 ↔ 𝑦 = 𝐵)
22	20, 21	bitri 277	. . . . . . 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 1844	. . . . 5 ⊢ (∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
25	16, 1	brco 5740	. . . . 5 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦 ∧ 𝑦◡2nd ⟨𝐴, 𝐵⟩))
26	5	clel3 3654	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑥 ∈ 𝑦))
27	24, 25, 26	3bitr4i 305	. . . 4 ⊢ (𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥 ∈ 𝐵)
28	19, 27	orbi12i 911	. . 3 ⊢ ((𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∨ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
29		brun 5116	. . 3 ⊢ (𝑥((◡1st ∘ E ) ∪ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(◡1st ∘ E )⟨𝐴, 𝐵⟩ ∨ 𝑥(◡2nd ∘ E )⟨𝐴, 𝐵⟩))
30		elun 4124	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
31	28, 29, 30	3bitr4ri 306	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥((◡1st ∘ E ) ∪ (◡2nd ∘ E ))⟨𝐴, 𝐵⟩)
32	1, 2, 3, 8, 31	brtxpsd3 33357	1 ⊢ (⟨𝐴, 𝐵⟩Cup𝐶 ↔ 𝐶 = (𝐴 ∪ 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494   ∪ cun 3933  ⟨cop 4572   class class class wbr 5065   E cep 5463   × cxp 5552  ^◡ccnv 5553   ∘ ccom 5558  1^st c1st 7686  2^nd c2nd 7687  Cupccup 33307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-symdif 4218  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-eprel 5464  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-1st 7688  df-2nd 7689  df-txp 33315  df-cup 33330

This theorem is referenced by:  brsuccf  33402