Theorem nfima 5076

Description: Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypotheses

Ref	Expression
nfima.1	|- F/_ x A
nfima.2	|- F/_ x B

Assertion

Ref	Expression
nfima	|- F/_ x ( A " B )

Proof of Theorem nfima

Step	Hyp	Ref	Expression
1		df-ima 4732	. 2 |- ( A " B ) = ran ( A |` B )
2		nfima.1	. . . 4 |- F/_ x A
3		nfima.2	. . . 4 |- F/_ x B
4	2, 3	nfres 5007	. . 3 |- F/_ x ( A |` B )
5	4	nfrn 4969	. 2 |- F/_ x ran ( A |` B )
6	1, 5	nfcxfr 2369	1 |- F/_ x ( A " B )

Syntax hints:   F/_wnfc 2359   ran crn 4720   |` cres 4721   "cima 4722

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-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  nfimad  5077  csbima12g  5089