Theorem funfvdm2f 5561

Description: The value of a function. Version of funfvdm2 5560 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 5560	. 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 2312	. . . . 5 |- F/_ y w
5	2, 3, 4	nfbr 4035	. . . 4 |- F/ y A F w
6		nfv 1521	. . . 4 |- F/ w A F y
7		breq2 3993	. . . 4 |- ( w = y -> ( A F w <-> A F y ) )
8	5, 6, 7	cbvab 2294	. . 3 |- { w | A F w } = { y | A F y }
9	8	unieqi 3806	. 2 |- U. { w | A F w } = U. { y | A F y }
10	1, 9	eqtrdi 2219	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. { y | A F y } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   F/_wnfc 2299   U.cuni 3796   class class class wbr 3989   dom cdm 4611   Fun wfun 5192   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)