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Theorem brbigcup 32342
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 32341 . . 3 Rel Bigcup
21brrelexi 5297 . 2 (𝐴 Bigcup 𝐵𝐴 ∈ V)
3 brbigcup.1 . . . 4 𝐵 ∈ V
4 eleq1 2838 . . . 4 ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 248 . . 3 ( 𝐴 = 𝐵 𝐴 ∈ V)
6 uniexb 7124 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
75, 6sylibr 224 . 2 ( 𝐴 = 𝐵𝐴 ∈ V)
8 breq1 4790 . . 3 (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵𝐴 Bigcup 𝐵))
9 unieq 4583 . . . 4 (𝑥 = 𝐴 𝑥 = 𝐴)
109eqeq1d 2773 . . 3 (𝑥 = 𝐴 → ( 𝑥 = 𝐵 𝐴 = 𝐵))
11 vex 3354 . . . . 5 𝑥 ∈ V
12 df-bigcup 32302 . . . . 5 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
13 brxp 5286 . . . . . 6 (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V))
1411, 3, 13mpbir2an 690 . . . . 5 𝑥(V × V)𝐵
15 epel 5166 . . . . . . 7 (𝑦 E 𝑧𝑦𝑧)
1615rexbii 3189 . . . . . 6 (∃𝑧𝑥 𝑦 E 𝑧 ↔ ∃𝑧𝑥 𝑦𝑧)
17 vex 3354 . . . . . . 7 𝑦 ∈ V
1817, 11coep 31979 . . . . . 6 (𝑦( E ∘ E )𝑥 ↔ ∃𝑧𝑥 𝑦 E 𝑧)
19 eluni2 4579 . . . . . 6 (𝑦 𝑥 ↔ ∃𝑧𝑥 𝑦𝑧)
2016, 18, 193bitr4ri 293 . . . . 5 (𝑦 𝑥𝑦( E ∘ E )𝑥)
2111, 3, 12, 14, 20brtxpsd3 32340 . . . 4 (𝑥 Bigcup 𝐵𝐵 = 𝑥)
22 eqcom 2778 . . . 4 (𝐵 = 𝑥 𝑥 = 𝐵)
2321, 22bitri 264 . . 3 (𝑥 Bigcup 𝐵 𝑥 = 𝐵)
248, 10, 23vtoclbg 3418 . 2 (𝐴 ∈ V → (𝐴 Bigcup 𝐵 𝐴 = 𝐵))
252, 7, 24pm5.21nii 367 1 (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351   cuni 4575   class class class wbr 4787   E cep 5162   × cxp 5248  ccom 5254   Bigcup cbigcup 32278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3994  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-eprel 5163  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038  df-1st 7319  df-2nd 7320  df-txp 32298  df-bigcup 32302
This theorem is referenced by:  dfbigcup2  32343  fvbigcup  32346  ellimits  32354  brapply  32382  dfrdg4  32395
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