Theorem nffvmpt1 5610

Description: Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)

Assertion

Ref	Expression
nffvmpt1	|- F/_ x ( ( x e. A |-> B ) ` C )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem nffvmpt1

Step	Hyp	Ref	Expression
1		nfmpt1 4153	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2350	. 2 |- F/_ x C
3	1, 2	nffv 5609	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2337   |-> cmpt 4121   ` cfv 5290

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 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-iota 5251  df-fv 5298

This theorem is referenced by:  fvmptt  5694  fmptco  5769  offval2  6197  ofrfval2  6198  mptelixpg  6844  dom2lem  6886  cc2  7414  fsumf1o  11816  fsum3cvg2  11820  fsumadd  11832  isummulc2  11852  isumshft  11916  fprodf1o  12014  prdsbas3  13234  txcnp  14858  cnmpt1t  14872  elplyd  15328