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Theorem brbigcup 33352
 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.
StepHypRef Expression
1 relbigcup 33351 . . 3 Rel Bigcup
21brrelex1i 5601 . 2 (𝐴 Bigcup 𝐵𝐴 ∈ V)
3 brbigcup.1 . . . 4 𝐵 ∈ V
4 eleq1 2898 . . . 4 ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 260 . . 3 ( 𝐴 = 𝐵 𝐴 ∈ V)
6 uniexb 7478 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
75, 6sylibr 236 . 2 ( 𝐴 = 𝐵𝐴 ∈ V)
8 breq1 5060 . . 3 (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵𝐴 Bigcup 𝐵))
9 unieq 4838 . . . 4 (𝑥 = 𝐴 𝑥 = 𝐴)
109eqeq1d 2821 . . 3 (𝑥 = 𝐴 → ( 𝑥 = 𝐵 𝐴 = 𝐵))
11 vex 3496 . . . . 5 𝑥 ∈ V
12 df-bigcup 33312 . . . . 5 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
13 brxp 5594 . . . . . 6 (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V))
1411, 3, 13mpbir2an 709 . . . . 5 𝑥(V × V)𝐵
15 epel 5462 . . . . . . 7 (𝑦 E 𝑧𝑦𝑧)
1615rexbii 3245 . . . . . 6 (∃𝑧𝑥 𝑦 E 𝑧 ↔ ∃𝑧𝑥 𝑦𝑧)
17 vex 3496 . . . . . . 7 𝑦 ∈ V
1817, 11coep 32980 . . . . . 6 (𝑦( E ∘ E )𝑥 ↔ ∃𝑧𝑥 𝑦 E 𝑧)
19 eluni2 4834 . . . . . 6 (𝑦 𝑥 ↔ ∃𝑧𝑥 𝑦𝑧)
2016, 18, 193bitr4ri 306 . . . . 5 (𝑦 𝑥𝑦( E ∘ E )𝑥)
2111, 3, 12, 14, 20brtxpsd3 33350 . . . 4 (𝑥 Bigcup 𝐵𝐵 = 𝑥)
22 eqcom 2826 . . . 4 (𝐵 = 𝑥 𝑥 = 𝐵)
2321, 22bitri 277 . . 3 (𝑥 Bigcup 𝐵 𝑥 = 𝐵)
248, 10, 23vtoclbg 3567 . 2 (𝐴 ∈ V → (𝐴 Bigcup 𝐵 𝐴 = 𝐵))
252, 7, 24pm5.21nii 382 1 (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1531   ∈ wcel 2108  ∃wrex 3137  Vcvv 3493  ∪ cuni 4830   class class class wbr 5057   E cep 5457   × cxp 5546   ∘ ccom 5552   Bigcup cbigcup 33288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-symdif 4217  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7681  df-2nd 7682  df-txp 33308  df-bigcup 33312 This theorem is referenced by:  dfbigcup2  33353  fvbigcup  33356  ellimits  33364  brapply  33392  dfrdg4  33405
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