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Theorem funfv2f 5784
Description: The value of a function. Version of funfv2 5783 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 5783 . 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 2571 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 4248 . . . 4  |-  F/ y  A F w
6 nfv 1629 . . . 4  |-  F/ w  A F y
7 breq2 4208 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2553 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 4017 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2483 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   {cab 2421   F/_wnfc 2558   U.cuni 4007   class class class wbr 4204   Fun wfun 5440   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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