Theorem nfima 6067

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 5689	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2		nfima.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfima.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfres 5983	. . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)
5	4	nfrn 5951	. 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵)
6	1, 5	nfcxfr 2901	1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵)

Syntax hints:  Ⅎwnfc 2883  ran crn 5677   ↾ cres 5678   “ cima 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by:  nfimad  6068  csbima12  6078  nfpred  6305  nfsup  9445  nfoi  9508  nfseq  13975  gsum2d2  19841  ptbasfi  23084  mbfposr  25168  itg1climres  25231  limciun  25410  funimass4f  31856  poimirlem16  36499  poimirlem19  36502  aomclem8  41793  areaquad  41955  nfcoll  43005  binomcxplemdvbinom  43102  binomcxplemdvsum  43104  binomcxplemnotnn0  43105  rfcnpre1  43693  rfcnpre2  43705  smfpimcc  45514