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Theorem funfv2f 5950
Description: The value of a function. Version of funfv2 5949 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 5949 . 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 2591 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 4470 . . . 4  |-  F/ y  A F w
6 nfv 1754 . . . 4  |-  F/ w  A F y
7 breq2 4430 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2570 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 4231 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2486 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {cab 2414   F/_wnfc 2577   U.cuni 4222   class class class wbr 4426   Fun wfun 5595   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609
This theorem is referenced by: (None)
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