Theorem nffv 5540

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 5239	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2332	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4064	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5197	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2329	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2319   class class class wbr 4018  ℩cio 5191  ‘cfv 5231

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239

This theorem is referenced by:  nffvmpt1  5541  nffvd  5542  dffn5imf  5587  fvmptssdm  5616  fvmptf  5624  eqfnfv2f  5633  ralrnmpt  5674  rexrnmpt  5675  ffnfvf  5691  funiunfvdmf  5781  dff13f  5787  nfiso  5823  nfrecs  6326  nffrec  6415  cc2  7284  nfseq  10473  seq3f1olemstep  10519  seq3f1olemp  10520  nfsum1  11382  nfsum  11383  fsumrelem  11497  nfcprod1  11580  nfcprod  11581  ctiunctlemfo  12458  ctiunct  12459  cnmpt11  14180  cnmpt21  14188  lgseisenlem2  14848