Theorem nfima 5049

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 4706	. 2 |- ( A " B ) = ran ( A |` B )
2		nfima.1	. . . 4 |- F/_ x A
3		nfima.2	. . . 4 |- F/_ x B
4	2, 3	nfres 4980	. . 3 |- F/_ x ( A |` B )
5	4	nfrn 4942	. 2 |- F/_ x ran ( A |` B )
6	1, 5	nfcxfr 2347	1 |- F/_ x ( A " B )

Syntax hints:   F/_wnfc 2337   ran crn 4694   |` cres 4695   "cima 4696

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-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-un 3178  df-in 3180  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  nfimad  5050  csbima12g  5062