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Theorem funfv2f 5724
Description: The value of a function. Version of funfv2 5723 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 5723 . 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 2516 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 4190 . . . 4  |-  F/ y  A F w
6 nfv 1626 . . . 4  |-  F/ w  A F y
7 breq2 4150 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2498 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 3960 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2428 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   {cab 2366   F/_wnfc 2503   U.cuni 3950   class class class wbr 4146   Fun wfun 5381   ` cfv 5387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-fv 5395
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