Theorem nfima 4897

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 4560	. 2 |- ( A " B ) = ran ( A |` B )
2		nfima.1	. . . 4 |- F/_ x A
3		nfima.2	. . . 4 |- F/_ x B
4	2, 3	nfres 4829	. . 3 |- F/_ x ( A |` B )
5	4	nfrn 4792	. 2 |- F/_ x ran ( A |` B )
6	1, 5	nfcxfr 2279	1 |- F/_ x ( A " B )

Syntax hints:   F/_wnfc 2269   ran crn 4548   |` cres 4549   "cima 4550

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-un 3080  df-in 3082  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by:  nfimad  4898  csbima12g  4908