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Theorem nfima 6025
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 𝑥𝐴
nfima.2 𝑥𝐵
Assertion
Ref Expression
nfima 𝑥(𝐴𝐵)

Proof of Theorem nfima
StepHypRef Expression
1 df-ima 5650 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 nfima.1 . . . 4 𝑥𝐴
3 nfima.2 . . . 4 𝑥𝐵
42, 3nfres 5943 . . 3 𝑥(𝐴𝐵)
54nfrn 5911 . 2 𝑥ran (𝐴𝐵)
61, 5nfcxfr 2902 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2884  ran crn 5638  cres 5639  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  nfimad  6026  csbima12  6035  nfpred  6262  nfsup  9395  nfoi  9458  nfseq  13925  gsum2d2  19759  ptbasfi  22955  mbfposr  25039  itg1climres  25102  limciun  25281  funimass4f  31604  poimirlem16  36144  poimirlem19  36147  aomclem8  41435  areaquad  41597  nfcoll  42628  binomcxplemdvbinom  42725  binomcxplemdvsum  42727  binomcxplemnotnn0  42728  rfcnpre1  43316  rfcnpre2  43328  smfpimcc  45139
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