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Theorem rabfmpunirn 27151
Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
Hypotheses
Ref Expression
rabfmpunirn.1  |-  F  =  ( x  e.  V  |->  { y  e.  W  |  ph } )
rabfmpunirn.2  |-  W  e. 
_V
rabfmpunirn.3  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rabfmpunirn  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  ( B  e.  W  /\  ps ) )
Distinct variable groups:    x, y, B    x, F, y    x, V, y    y, W    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    W( x)

Proof of Theorem rabfmpunirn
StepHypRef Expression
1 rabfmpunirn.1 . . . 4  |-  F  =  ( x  e.  V  |->  { y  e.  W  |  ph } )
2 df-rab 2818 . . . . 5  |-  { y  e.  W  |  ph }  =  { y  |  ( y  e.  W  /\  ph ) }
32mpteq2i 4525 . . . 4  |-  ( x  e.  V  |->  { y  e.  W  |  ph } )  =  ( x  e.  V  |->  { y  |  ( y  e.  W  /\  ph ) } )
41, 3eqtri 2491 . . 3  |-  F  =  ( x  e.  V  |->  { y  |  ( y  e.  W  /\  ph ) } )
5 rabfmpunirn.2 . . . 4  |-  W  e. 
_V
65zfausab 4591 . . 3  |-  { y  |  ( y  e.  W  /\  ph ) }  e.  _V
7 eleq1 2534 . . . 4  |-  ( y  =  B  ->  (
y  e.  W  <->  B  e.  W ) )
8 rabfmpunirn.3 . . . 4  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
97, 8anbi12d 710 . . 3  |-  ( y  =  B  ->  (
( y  e.  W  /\  ph )  <->  ( B  e.  W  /\  ps )
) )
104, 6, 9abfmpunirn 27150 . 2  |-  ( B  e.  U. ran  F  <->  ( B  e.  _V  /\  E. x  e.  V  ( B  e.  W  /\  ps ) ) )
11 elex 3117 . . . . 5  |-  ( B  e.  W  ->  B  e.  _V )
1211adantr 465 . . . 4  |-  ( ( B  e.  W  /\  ps )  ->  B  e. 
_V )
1312rexlimivw 2947 . . 3  |-  ( E. x  e.  V  ( B  e.  W  /\  ps )  ->  B  e. 
_V )
1413pm4.71ri 633 . 2  |-  ( E. x  e.  V  ( B  e.  W  /\  ps )  <->  ( B  e. 
_V  /\  E. x  e.  V  ( B  e.  W  /\  ps )
) )
1510, 14bitr4i 252 1  |-  ( B  e.  U. ran  F  <->  E. x  e.  V  ( B  e.  W  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   E.wrex 2810   {crab 2813   _Vcvv 3108   U.cuni 4240    |-> cmpt 4500   ran crn 4995
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
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-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  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-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589
This theorem is referenced by: (None)
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