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Theorem nfima 6068
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 5690 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 nfima.1 . . . 4 𝑥𝐴
3 nfima.2 . . . 4 𝑥𝐵
42, 3nfres 5984 . . 3 𝑥(𝐴𝐵)
54nfrn 5952 . 2 𝑥ran (𝐴𝐵)
61, 5nfcxfr 2902 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2884  ran crn 5678  cres 5679  cima 5680
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 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  nfimad  6069  csbima12  6079  nfpred  6306  nfsup  9446  nfoi  9509  nfseq  13976  gsum2d2  19842  ptbasfi  23085  mbfposr  25169  itg1climres  25232  limciun  25411  funimass4f  31861  poimirlem16  36504  poimirlem19  36507  aomclem8  41803  areaquad  41965  nfcoll  43015  binomcxplemdvbinom  43112  binomcxplemdvsum  43114  binomcxplemnotnn0  43115  rfcnpre1  43703  rfcnpre2  43715  smfpimcc  45524
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