Theorem nfima 5090

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 4744	. 2 |- ( A " B ) = ran ( A |` B )
2		nfima.1	. . . 4 |- F/_ x A
3		nfima.2	. . . 4 |- F/_ x B
4	2, 3	nfres 5021	. . 3 |- F/_ x ( A |` B )
5	4	nfrn 4983	. 2 |- F/_ x ran ( A |` B )
6	1, 5	nfcxfr 2372	1 |- F/_ x ( A " B )

Syntax hints:   F/_wnfc 2362   ran crn 4732   |` cres 4733   "cima 4734

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  nfimad  5091  csbima12g  5104