Theorem brbigcup 35940

Description: Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.)

Hypothesis

Ref	Expression
brbigcup.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brbigcup	⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)

Proof of Theorem brbigcup

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

Step	Hyp	Ref	Expression
1		relbigcup 35939	. . 3 ⊢ Rel Bigcup
2	1	brrelex1i 5670	. 2 ⊢ (𝐴 Bigcup 𝐵 → 𝐴 ∈ V)
3		brbigcup.1	. . . 4 ⊢ 𝐵 ∈ V
4		eleq1 2819	. . . 4 ⊢ (∪ 𝐴 = 𝐵 → (∪ 𝐴 ∈ V ↔ 𝐵 ∈ V))
5	3, 4	mpbiri 258	. . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ∈ V)
6		uniexb 7697	. . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
7	5, 6	sylibr 234	. 2 ⊢ (∪ 𝐴 = 𝐵 → 𝐴 ∈ V)
8		breq1 5092	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵 ↔ 𝐴 Bigcup 𝐵))
9		unieq 4867	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
10	9	eqeq1d 2733	. . 3 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐵 ↔ ∪ 𝐴 = 𝐵))
11		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
12		df-bigcup 35900	. . . . 5 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
13		brxp 5663	. . . . . 6 ⊢ (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V))
14	11, 3, 13	mpbir2an 711	. . . . 5 ⊢ 𝑥(V × V)𝐵
15		epel 5517	. . . . . . 7 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
16	15	rexbii 3079	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑥 𝑦 E 𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧)
17		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
18	17, 11	coep 35796	. . . . . 6 ⊢ (𝑦( E ∘ E )𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 E 𝑧)
19		eluni2 4860	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧)
20	16, 18, 19	3bitr4ri 304	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ 𝑦( E ∘ E )𝑥)
21	11, 3, 12, 14, 20	brtxpsd3 35938	. . . 4 ⊢ (𝑥 Bigcup 𝐵 ↔ 𝐵 = ∪ 𝑥)
22		eqcom 2738	. . . 4 ⊢ (𝐵 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝐵)
23	21, 22	bitri 275	. . 3 ⊢ (𝑥 Bigcup 𝐵 ↔ ∪ 𝑥 = 𝐵)
24	8, 10, 23	vtoclbg 3510	. 2 ⊢ (𝐴 ∈ V → (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵))
25	2, 7, 24	pm5.21nii 378	1 ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436  ∪ cuni 4856   class class class wbr 5089   E cep 5513   × cxp 5612   ∘ ccom 5618   Bigcup cbigcup 35876

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-symdif 4200  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-eprel 5514  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921  df-2nd 7922  df-txp 35896  df-bigcup 35900

This theorem is referenced by:  dfbigcup2  35941  fvbigcup  35944  ellimits  35952  brapply  35980  dfrdg4  35995