Theorem nffv 5645

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 5332	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2372	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4133	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5288	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2369	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2359   class class class wbr 4086  ℩cio 5282  ‘cfv 5324

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332

This theorem is referenced by:  nffvmpt1  5646  nffvd  5647  dffn5imf  5697  fvmptssdm  5727  fvmptf  5735  eqfnfv2f  5744  ralrnmpt  5785  rexrnmpt  5786  ffnfvf  5802  funiunfvdmf  5900  dff13f  5906  nfiso  5942  nfrecs  6468  nffrec  6557  cc2  7476  nfseq  10709  seq3f1olemstep  10766  seq3f1olemp  10767  nfsum1  11907  nfsum  11908  fsumrelem  12022  nfcprod1  12105  nfcprod  12106  ctiunctlemfo  13050  ctiunct  13051  prdsbas3  13360  cnmpt11  14997  cnmpt21  15005  lgseisenlem2  15790