Theorem nffv 5708

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 5388	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2392	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4178	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaw 5344	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2389	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

This proof depends on syntax axioms:  Ⅎwnfc 2379   class class class wbr 4131  ℩cio 5338  ‘cfv 5380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-iota 5340  df-fv 5388

This theorem is used by:  nffvmpt1  5709  nffvd  5710  dffn5imf  5761  fvmptssdm  5793  fvmptf  5801  eqfnfv2f  5813  ralrnmpt  5853  rexrnmpt  5854  ffnfvf  5870  dfimafnf  5958  funiunfvdmf  5973  dff13f  5979  nfiso  6015  nfrecs  6582  nffrec  6671  cc2  7639  nfseq  10915  seq3f1olemstep  10972  seq3f1olemp  10973  nfsum1  12147  nfsum  12148  fsumrelem  12263  nfcprod1  12346  nfcprod  12347  ctiunctlemfo  13388  ctiunct  13389  prdsbas3  14294  cnmpt11  15509  cnmpt21  15517  lgseisenlem2  16395