Theorem nfima 5936

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 5567	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2		nfima.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfima.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfres 5854	. . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)
5	4	nfrn 5823	. 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵)
6	1, 5	nfcxfr 2975	1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵)

Syntax hints:  Ⅎwnfc 2961  ran crn 5555   ↾ cres 5556   “ cima 5557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567

This theorem is referenced by:  nfimad  5937  csbima12  5946  nfpred  6152  nfsup  8914  nfoi  8977  nfseq  13378  gsum2d2  19093  ptbasfi  22188  mbfposr  24252  itg1climres  24314  limciun  24491  funimass4f  30381  poimirlem16  34907  poimirlem19  34910  aomclem8  39659  areaquad  39821  nfcoll  40590  binomcxplemdvbinom  40683  binomcxplemdvsum  40685  binomcxplemnotnn0  40686  rfcnpre1  41274  rfcnpre2  41286  smfpimcc  43081