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Theorem funfv2f 5937
Description: The value of a function. Version of funfv2 5936 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
Hypotheses
Ref Expression
funfv2f.1  |-  F/_ y A
funfv2f.2  |-  F/_ y F
Assertion
Ref Expression
funfv2f  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )

Proof of Theorem funfv2f
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 funfv2 5936 . 2  |-  ( Fun 
F  ->  ( F `  A )  =  U. { w  |  A F w } )
2 funfv2f.1 . . . . 5  |-  F/_ y A
3 funfv2f.2 . . . . 5  |-  F/_ y F
4 nfcv 2629 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 4491 . . . 4  |-  F/ y  A F w
6 nfv 1683 . . . 4  |-  F/ w  A F y
7 breq2 4451 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2608 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 4254 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2524 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   {cab 2452   F/_wnfc 2615   U.cuni 4245   class class class wbr 4447   Fun wfun 5582   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by: (None)
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