MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfima Structured version   Visualization version   GIF version

Theorem nfima 5438
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 5092 . 2 (𝐴𝐵) = ran (𝐴𝐵)
2 nfima.1 . . . 4 𝑥𝐴
3 nfima.2 . . . 4 𝑥𝐵
42, 3nfres 5363 . . 3 𝑥(𝐴𝐵)
54nfrn 5333 . 2 𝑥ran (𝐴𝐵)
61, 5nfcxfr 2759 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2748  ran crn 5080  cres 5081  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by:  nfimad  5439  csbima12  5447  nfpred  5649  nfsup  8309  nfoi  8371  nfseq  12759  gsum2d2  18305  ptbasfi  21307  mbfposr  23342  itg1climres  23404  limciun  23581  funimass4f  29303  poimirlem16  33092  poimirlem19  33095  aomclem8  37146  areaquad  37318  binomcxplemdvbinom  38069  binomcxplemdvsum  38071  binomcxplemnotnn0  38072  rfcnpre1  38696  rfcnpre2  38708  smfpimcc  40347
  Copyright terms: Public domain W3C validator