Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brbigcup Structured version   Visualization version   GIF version

Theorem brbigcup 35899
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 35898 . . 3 Rel Bigcup
21brrelex1i 5741 . 2 (𝐴 Bigcup 𝐵𝐴 ∈ V)
3 brbigcup.1 . . . 4 𝐵 ∈ V
4 eleq1 2829 . . . 4 ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 258 . . 3 ( 𝐴 = 𝐵 𝐴 ∈ V)
6 uniexb 7784 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
75, 6sylibr 234 . 2 ( 𝐴 = 𝐵𝐴 ∈ V)
8 breq1 5146 . . 3 (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵𝐴 Bigcup 𝐵))
9 unieq 4918 . . . 4 (𝑥 = 𝐴 𝑥 = 𝐴)
109eqeq1d 2739 . . 3 (𝑥 = 𝐴 → ( 𝑥 = 𝐵 𝐴 = 𝐵))
11 vex 3484 . . . . 5 𝑥 ∈ V
12 df-bigcup 35859 . . . . 5 Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))
13 brxp 5734 . . . . . 6 (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V))
1411, 3, 13mpbir2an 711 . . . . 5 𝑥(V × V)𝐵
15 epel 5587 . . . . . . 7 (𝑦 E 𝑧𝑦𝑧)
1615rexbii 3094 . . . . . 6 (∃𝑧𝑥 𝑦 E 𝑧 ↔ ∃𝑧𝑥 𝑦𝑧)
17 vex 3484 . . . . . . 7 𝑦 ∈ V
1817, 11coep 35752 . . . . . 6 (𝑦( E ∘ E )𝑥 ↔ ∃𝑧𝑥 𝑦 E 𝑧)
19 eluni2 4911 . . . . . 6 (𝑦 𝑥 ↔ ∃𝑧𝑥 𝑦𝑧)
2016, 18, 193bitr4ri 304 . . . . 5 (𝑦 𝑥𝑦( E ∘ E )𝑥)
2111, 3, 12, 14, 20brtxpsd3 35897 . . . 4 (𝑥 Bigcup 𝐵𝐵 = 𝑥)
22 eqcom 2744 . . . 4 (𝐵 = 𝑥 𝑥 = 𝐵)
2321, 22bitri 275 . . 3 (𝑥 Bigcup 𝐵 𝑥 = 𝐵)
248, 10, 23vtoclbg 3557 . 2 (𝐴 ∈ V → (𝐴 Bigcup 𝐵 𝐴 = 𝐵))
252, 7, 24pm5.21nii 378 1 (𝐴 Bigcup 𝐵 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480   cuni 4907   class class class wbr 5143   E cep 5583   × cxp 5683  ccom 5689   Bigcup cbigcup 35835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-bigcup 35859
This theorem is referenced by:  dfbigcup2  35900  fvbigcup  35903  ellimits  35911  brapply  35939  dfrdg4  35952
  Copyright terms: Public domain W3C validator