Theorem nffv 5658

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 5341	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2375	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4140	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5297	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2372	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2362   class class class wbr 4093  ℩cio 5291  ‘cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341

This theorem is referenced by:  nffvmpt1  5659  nffvd  5660  dffn5imf  5710  fvmptssdm  5740  fvmptf  5748  eqfnfv2f  5757  ralrnmpt  5797  rexrnmpt  5798  ffnfvf  5814  funiunfvdmf  5915  dff13f  5921  nfiso  5957  nfrecs  6516  nffrec  6605  cc2  7529  nfseq  10765  seq3f1olemstep  10822  seq3f1olemp  10823  nfsum1  11979  nfsum  11980  fsumrelem  12095  nfcprod1  12178  nfcprod  12179  ctiunctlemfo  13123  ctiunct  13124  prdsbas3  13433  cnmpt11  15077  cnmpt21  15085  lgseisenlem2  15873