Theorem nffv 5639

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	⊢ Ⅎ𝑥𝐹
nffv.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nffv	⊢ Ⅎ𝑥(𝐹‘𝐴)

Proof of Theorem nffv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5326	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2372	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4130	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5282	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2369	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2359   class class class wbr 4083  ℩cio 5276  ‘cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326

This theorem is referenced by:  nffvmpt1  5640  nffvd  5641  dffn5imf  5691  fvmptssdm  5721  fvmptf  5729  eqfnfv2f  5738  ralrnmpt  5779  rexrnmpt  5780  ffnfvf  5796  funiunfvdmf  5894  dff13f  5900  nfiso  5936  nfrecs  6459  nffrec  6548  cc2  7464  nfseq  10691  seq3f1olemstep  10748  seq3f1olemp  10749  nfsum1  11882  nfsum  11883  fsumrelem  11997  nfcprod1  12080  nfcprod  12081  ctiunctlemfo  13025  ctiunct  13026  prdsbas3  13335  cnmpt11  14972  cnmpt21  14980  lgseisenlem2  15765