Theorem nfima 6086

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	⊢ Ⅎ𝑥𝐴
nfima.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfima	⊢ Ⅎ𝑥(𝐴 “ 𝐵)

Proof of Theorem nfima

Step	Hyp	Ref	Expression
1		df-ima 5698	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2		nfima.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfima.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfres 5999	. . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)
5	4	nfrn 5963	. 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵)
6	1, 5	nfcxfr 2903	1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵)

Syntax hints:  Ⅎwnfc 2890  ran crn 5686   ↾ cres 5687   “ cima 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  nfimad  6087  csbima12  6097  nfpred  6326  nfsup  9491  nfoi  9554  nfseq  14052  gsum2d2  19992  ptbasfi  23589  mbfposr  25687  itg1climres  25749  limciun  25929  nfseqs  28293  funimass4f  32647  poimirlem16  37643  poimirlem19  37646  aomclem8  43073  areaquad  43228  nfcoll  44275  binomcxplemdvbinom  44372  binomcxplemdvsum  44374  binomcxplemnotnn0  44375  rfcnpre1  45024  rfcnpre2  45036  smfpimcc  46823