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Theorem brbigcup 29113
Description: Binary relationship over  Bigcup. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
brbigcup.1  |-  B  e. 
_V
Assertion
Ref Expression
brbigcup  |-  ( A
Bigcup B  <->  U. A  =  B )

Proof of Theorem brbigcup
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relbigcup 29112 . . 3  |-  Rel  Bigcup
21brrelexi 5034 . 2  |-  ( A
Bigcup B  ->  A  e.  _V )
3 brbigcup.1 . . . 4  |-  B  e. 
_V
4 eleq1 2534 . . . 4  |-  ( U. A  =  B  ->  ( U. A  e.  _V  <->  B  e.  _V ) )
53, 4mpbiri 233 . . 3  |-  ( U. A  =  B  ->  U. A  e.  _V )
6 uniexb 6583 . . 3  |-  ( A  e.  _V  <->  U. A  e. 
_V )
75, 6sylibr 212 . 2  |-  ( U. A  =  B  ->  A  e.  _V )
8 breq1 4445 . . 3  |-  ( x  =  A  ->  (
x Bigcup B  <->  A Bigcup B ) )
9 unieq 4248 . . . 4  |-  ( x  =  A  ->  U. x  =  U. A )
109eqeq1d 2464 . . 3  |-  ( x  =  A  ->  ( U. x  =  B  <->  U. A  =  B ) )
11 vex 3111 . . . . 5  |-  x  e. 
_V
12 df-bigcup 29072 . . . . 5  |-  Bigcup  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  _E  )  (x)  _V )
) )
13 brxp 5024 . . . . . 6  |-  ( x ( _V  X.  _V ) B  <->  ( x  e. 
_V  /\  B  e.  _V ) )
1411, 3, 13mpbir2an 913 . . . . 5  |-  x ( _V  X.  _V ) B
15 epel 4789 . . . . . . 7  |-  ( y  _E  z  <->  y  e.  z )
1615rexbii 2960 . . . . . 6  |-  ( E. z  e.  x  y  _E  z  <->  E. z  e.  x  y  e.  z )
17 vex 3111 . . . . . . 7  |-  y  e. 
_V
1817, 11coep 28745 . . . . . 6  |-  ( y (  _E  o.  _E  ) x  <->  E. z  e.  x  y  _E  z )
19 eluni2 4244 . . . . . 6  |-  ( y  e.  U. x  <->  E. z  e.  x  y  e.  z )
2016, 18, 193bitr4ri 278 . . . . 5  |-  ( y  e.  U. x  <->  y (  _E  o.  _E  ) x )
2111, 3, 12, 14, 20brtxpsd3 29111 . . . 4  |-  ( x
Bigcup B  <->  B  =  U. x )
22 eqcom 2471 . . . 4  |-  ( B  =  U. x  <->  U. x  =  B )
2321, 22bitri 249 . . 3  |-  ( x
Bigcup B  <->  U. x  =  B )
248, 10, 23vtoclbg 3167 . 2  |-  ( A  e.  _V  ->  ( A Bigcup B  <->  U. A  =  B ) )
252, 7, 24pm5.21nii 353 1  |-  ( A
Bigcup B  <->  U. A  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374    e. wcel 1762   E.wrex 2810   _Vcvv 3108   U.cuni 4240   class class class wbr 4442    _E cep 4784    X. cxp 4992    o. ccom 4998   Bigcupcbigcup 29048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-eprel 4786  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6776  df-2nd 6777  df-symdif 29033  df-txp 29068  df-bigcup 29072
This theorem is referenced by:  dfbigcup2  29114  fvbigcup  29117  ellimits  29125  brapply  29153  dfrdg4  29165
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