Theorem nffv 5599

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 5288	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2349	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4098	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5245	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2346	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2336   class class class wbr 4051  ℩cio 5239  ‘cfv 5280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288

This theorem is referenced by:  nffvmpt1  5600  nffvd  5601  dffn5imf  5647  fvmptssdm  5677  fvmptf  5685  eqfnfv2f  5694  ralrnmpt  5735  rexrnmpt  5736  ffnfvf  5752  funiunfvdmf  5846  dff13f  5852  nfiso  5888  nfrecs  6406  nffrec  6495  cc2  7399  nfseq  10624  seq3f1olemstep  10681  seq3f1olemp  10682  nfsum1  11742  nfsum  11743  fsumrelem  11857  nfcprod1  11940  nfcprod  11941  ctiunctlemfo  12885  ctiunct  12886  prdsbas3  13194  cnmpt11  14830  cnmpt21  14838  lgseisenlem2  15623