Theorem nfima 6052

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 5656	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2		nfima.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfima.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfres 5963	. . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)
5	4	nfrn 5924	. 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵)
6	1, 5	nfcxfr 2921	1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵)

Syntax hints:  Ⅎwnfc 2908  ran crn 5644   ↾ cres 5645   “ cima 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656

This theorem is referenced by:  nfimad  6053  csbima12  6063  nfpred  6287  nfsup  9390  nfoi  9455  nfseq  14017  gsum2d2  20004  ptbasfi  23628  mbfposr  25701  itg1climres  25763  limciun  25943  nfseqs  28367  funimass4f  32799  poimirlem16  38095  poimirlem19  38098  aomclem8  43598  areaquad  43753  nfcoll  44792  binomcxplemdvbinom  44889  binomcxplemdvsum  44891  binomcxplemnotnn0  44892  rfcnpre1  45559  rfcnpre2  45571  smfpimcc  47342