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Theorem rabfmpunirn 27815
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 2760 . . . . 5  |-  { y  e.  W  |  ph }  =  { y  |  ( y  e.  W  /\  ph ) }
32mpteq2i 4475 . . . 4  |-  ( x  e.  V  |->  { y  e.  W  |  ph } )  =  ( x  e.  V  |->  { y  |  ( y  e.  W  /\  ph ) } )
41, 3eqtri 2429 . . 3  |-  F  =  ( x  e.  V  |->  { y  |  ( y  e.  W  /\  ph ) } )
5 rabfmpunirn.2 . . . 4  |-  W  e. 
_V
65zfausab 4540 . . 3  |-  { y  |  ( y  e.  W  /\  ph ) }  e.  _V
7 eleq1 2472 . . . 4  |-  ( y  =  B  ->  (
y  e.  W  <->  B  e.  W ) )
8 rabfmpunirn.3 . . . 4  |-  ( y  =  B  ->  ( ph 
<->  ps ) )
97, 8anbi12d 709 . . 3  |-  ( y  =  B  ->  (
( y  e.  W  /\  ph )  <->  ( B  e.  W  /\  ps )
) )
104, 6, 9abfmpunirn 27814 . 2  |-  ( B  e.  U. ran  F  <->  ( B  e.  _V  /\  E. x  e.  V  ( B  e.  W  /\  ps ) ) )
11 elex 3065 . . . . 5  |-  ( B  e.  W  ->  B  e.  _V )
1211adantr 463 . . . 4  |-  ( ( B  e.  W  /\  ps )  ->  B  e. 
_V )
1312rexlimivw 2890 . . 3  |-  ( E. x  e.  V  ( B  e.  W  /\  ps )  ->  B  e. 
_V )
1413pm4.71ri 631 . 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 367    = wceq 1403    e. wcel 1840   {cab 2385   E.wrex 2752   {crab 2755   _Vcvv 3056   U.cuni 4188    |-> cmpt 4450   ran crn 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531
This theorem is referenced by: (None)
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