Theorem nffv 5185

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 4910	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		nffv.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nffv.1	. . . 4 ⊢ Ⅎ𝑥𝐹
4		nfcv 2178	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 3808	. . 3 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
6	5	nfiotaxy 4871	. 2 ⊢ Ⅎ𝑥(℩𝑦𝐴𝐹𝑦)
7	1, 6	nfcxfr 2175	1 ⊢ Ⅎ𝑥(𝐹‘𝐴)

Syntax hints:  Ⅎwnfc 2165   class class class wbr 3764  ℩cio 4865  ‘cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910

This theorem is referenced by:  nffvmpt1  5186  nffvd  5187  dffn5imf  5228  fvmptssdm  5255  fvmptf  5263  eqfnfv2f  5269  ralrnmpt  5309  rexrnmpt  5310  ffnfvf  5324  funiunfvdmf  5403  dff13f  5409  nfiso  5446  nfrecs  5922  nffrec  5982  nfiseq  9218  nfsum1  9875  nfsum  9876