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Theorem brbigcup 31682
Description: Binary relationship 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.
StepHypRef Expression
1 relbigcup 31681 . . 3 Rel Bigcup
21brrelexi 5123 . 2 (𝐴 Bigcup 𝐵𝐴 ∈ V)
3 brbigcup.1 . . . 4 𝐵 ∈ V
4 eleq1 2686 . . . 4 ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 248 . . 3 ( 𝐴 = 𝐵 𝐴 ∈ V)
6 uniexb 6928 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
75, 6sylibr 224 . 2 ( 𝐴 = 𝐵𝐴 ∈ V)
8 breq1 4621 . . 3 (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵𝐴 Bigcup 𝐵))
9 unieq 4415 . . . 4 (𝑥 = 𝐴 𝑥 = 𝐴)
109eqeq1d 2623 . . 3 (𝑥 = 𝐴 → ( 𝑥 = 𝐵 𝐴 = 𝐵))
11 vex 3192 . . . . 5 𝑥 ∈ V
12 df-bigcup 31641 . . . . 5 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
13 brxp 5112 . . . . . 6 (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V))
1411, 3, 13mpbir2an 954 . . . . 5 𝑥(V × V)𝐵
15 epel 4993 . . . . . . 7 (𝑦 E 𝑧𝑦𝑧)
1615rexbii 3035 . . . . . 6 (∃𝑧𝑥 𝑦 E 𝑧 ↔ ∃𝑧𝑥 𝑦𝑧)
17 vex 3192 . . . . . . 7 𝑦 ∈ V
1817, 11coep 31384 . . . . . 6 (𝑦( E ∘ E )𝑥 ↔ ∃𝑧𝑥 𝑦 E 𝑧)
19 eluni2 4411 . . . . . 6 (𝑦 𝑥 ↔ ∃𝑧𝑥 𝑦𝑧)
2016, 18, 193bitr4ri 293 . . . . 5 (𝑦 𝑥𝑦( E ∘ E )𝑥)
2111, 3, 12, 14, 20brtxpsd3 31680 . . . 4 (𝑥 Bigcup 𝐵𝐵 = 𝑥)
22 eqcom 2628 . . . 4 (𝐵 = 𝑥 𝑥 = 𝐵)
2321, 22bitri 264 . . 3 (𝑥 Bigcup 𝐵 𝑥 = 𝐵)
248, 10, 23vtoclbg 3256 . 2 (𝐴 ∈ V → (𝐴 Bigcup 𝐵 𝐴 = 𝐵))
252, 7, 24pm5.21nii 368 1 (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3189   cuni 4407   class class class wbr 4618   E cep 4988   × cxp 5077  ccom 5083   Bigcup cbigcup 31617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-symdif 3827  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-bigcup 31641
This theorem is referenced by:  dfbigcup2  31683  fvbigcup  31686  ellimits  31694  brapply  31722  dfrdg4  31735
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