Theorem nfima 6097

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 5713	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2		nfima.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfima.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfres 6011	. . 3 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵)
5	4	nfrn 5977	. 2 ⊢ Ⅎ𝑥ran (𝐴 ↾ 𝐵)
6	1, 5	nfcxfr 2906	1 ⊢ Ⅎ𝑥(𝐴 “ 𝐵)

Syntax hints:  Ⅎwnfc 2893  ran crn 5701   ↾ cres 5702   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  nfimad  6098  csbima12  6108  nfpred  6337  nfsup  9520  nfoi  9583  nfseq  14062  gsum2d2  20016  ptbasfi  23610  mbfposr  25706  itg1climres  25769  limciun  25949  nfseqs  28311  funimass4f  32656  poimirlem16  37596  poimirlem19  37599  aomclem8  43018  areaquad  43177  nfcoll  44225  binomcxplemdvbinom  44322  binomcxplemdvsum  44324  binomcxplemnotnn0  44325  rfcnpre1  44919  rfcnpre2  44931  smfpimcc  46729