Theorem nffv 5580

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 5276	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2347	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4089	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5233	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2344	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2334   class class class wbr 4043  ℩cio 5227  ‘cfv 5268

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5229  df-fv 5276

This theorem is referenced by:  nffvmpt1  5581  nffvd  5582  dffn5imf  5628  fvmptssdm  5658  fvmptf  5666  eqfnfv2f  5675  ralrnmpt  5716  rexrnmpt  5717  ffnfvf  5733  funiunfvdmf  5823  dff13f  5829  nfiso  5865  nfrecs  6383  nffrec  6472  cc2  7361  nfseq  10583  seq3f1olemstep  10640  seq3f1olemp  10641  nfsum1  11586  nfsum  11587  fsumrelem  11701  nfcprod1  11784  nfcprod  11785  ctiunctlemfo  12729  ctiunct  12730  prdsbas3  13037  cnmpt11  14673  cnmpt21  14681  lgseisenlem2  15466