Theorem funfvdm2f 5667

Description: The value of a function. Version of funfvdm2 5666 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 5666	. 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 2350	. . . . 5 |- F/_ y w
5	2, 3, 4	nfbr 4106	. . . 4 |- F/ y A F w
6		nfv 1552	. . . 4 |- F/ w A F y
7		breq2 4063	. . . 4 |- ( w = y -> ( A F w <-> A F y ) )
8	5, 6, 7	cbvab 2331	. . 3 |- { w | A F w } = { y | A F y }
9	8	unieqi 3874	. 2 |- U. { w | A F w } = U. { y | A F y }
10	1, 9	eqtrdi 2256	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   {cab 2193   F/_wnfc 2337   U.cuni 3864   class class class wbr 4059   dom cdm 4693   Fun wfun 5284   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by: (None)