Theorem funfvdm2f 5707

Description: The value of a function. Version of funfvdm2 5706 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 5706	. 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 2372	. . . . 5 |- F/_ y w
5	2, 3, 4	nfbr 4133	. . . 4 |- F/ y A F w
6		nfv 1574	. . . 4 |- F/ w A F y
7		breq2 4090	. . . 4 |- ( w = y -> ( A F w <-> A F y ) )
8	5, 6, 7	cbvab 2353	. . 3 |- { w | A F w } = { y | A F y }
9	8	unieqi 3901	. 2 |- U. { w | A F w } = U. { y | A F y }
10	1, 9	eqtrdi 2278	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   F/_wnfc 2359   U.cuni 3891   class class class wbr 4086   dom cdm 4723   Fun wfun 5318   ` cfv 5324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by: (None)