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Theorem nffv 5397
Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nffv.1 𝑥𝐹
nffv.2 𝑥𝐴
Assertion
Ref Expression
nffv 𝑥(𝐹𝐴)

Proof of Theorem nffv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-fv 5099 . 2 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 nffv.2 . . . 4 𝑥𝐴
3 nffv.1 . . . 4 𝑥𝐹
4 nfcv 2256 . . . 4 𝑥𝑦
52, 3, 4nfbr 3942 . . 3 𝑥 𝐴𝐹𝑦
65nfiotaxy 5060 . 2 𝑥(℩𝑦𝐴𝐹𝑦)
71, 6nfcxfr 2253 1 𝑥(𝐹𝐴)
Colors of variables: wff set class
Syntax hints:  wnfc 2243   class class class wbr 3897  cio 5054  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099
This theorem is referenced by:  nffvmpt1  5398  nffvd  5399  dffn5imf  5442  fvmptssdm  5471  fvmptf  5479  eqfnfv2f  5488  ralrnmpt  5528  rexrnmpt  5529  ffnfvf  5545  funiunfvdmf  5631  dff13f  5637  nfiso  5673  nfrecs  6170  nffrec  6259  nfseq  10179  seq3f1olemstep  10225  seq3f1olemp  10226  nfsum1  11076  nfsum  11077  fsumrelem  11191  ctiunctlemfo  11858  ctiunct  11859  cnmpt11  12358  cnmpt21  12366
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