Theorem funfvdm2f 5494

Description: The value of a function. Version of funfvdm2 5493 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Hypotheses

Ref	Expression
funfvdm2f.1	|- F/_ y A
funfvdm2f.2	|- F/_ y F

Assertion

Ref	Expression
funfvdm2f	|- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )

Proof of Theorem funfvdm2f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvdm2 5493	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { w | A F w } )
2		funfvdm2f.1	. . . . 5 |- F/_ y A
3		funfvdm2f.2	. . . . 5 |- F/_ y F
4		nfcv 2282	. . . . 5 |- F/_ y w
5	2, 3, 4	nfbr 3982	. . . 4 |- F/ y A F w
6		nfv 1509	. . . 4 |- F/ w A F y
7		breq2 3941	. . . 4 |- ( w = y -> ( A F w <-> A F y ) )
8	5, 6, 7	cbvab 2264	. . 3 |- { w | A F w } = { y | A F y }
9	8	unieqi 3754	. 2 |- U. { w | A F w } = U. { y | A F y }
10	1, 9	eqtrdi 2189	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {cab 2126   F/_wnfc 2269   U.cuni 3744   class class class wbr 3937   dom cdm 4547   Fun wfun 5125   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by: (None)