Theorem nffvmpt1 5683

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 4205	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2386	. 2 |- F/_ x C
3	1, 2	nffv 5682	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2373   |-> cmpt 4173   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-iota 5314  df-fv 5362

This theorem is referenced by:  fvmptt  5771  fmptco  5845  offval2  6284  ofrfval2  6285  mptelixpg  6971  dom2lem  7013  cc2  7583  fsumf1o  12080  fsum3cvg2  12084  fsumadd  12096  isummulc2  12116  isumshft  12180  fprodf1o  12278  prdsbas3  13517  txcnp  15153  cnmpt1t  15167  elplyd  15623