Theorem nffv 5526

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 5225	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2319	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4050	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5183	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2316	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2306   class class class wbr 4004  ℩cio 5177  ‘cfv 5217

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225

This theorem is referenced by:  nffvmpt1  5527  nffvd  5528  dffn5imf  5572  fvmptssdm  5601  fvmptf  5609  eqfnfv2f  5618  ralrnmpt  5659  rexrnmpt  5660  ffnfvf  5676  funiunfvdmf  5765  dff13f  5771  nfiso  5807  nfrecs  6308  nffrec  6397  cc2  7266  nfseq  10455  seq3f1olemstep  10501  seq3f1olemp  10502  nfsum1  11364  nfsum  11365  fsumrelem  11479  nfcprod1  11562  nfcprod  11563  ctiunctlemfo  12440  ctiunct  12441  cnmpt11  13786  cnmpt21  13794  lgseisenlem2  14454