Theorem nffvmpt1 5686

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 4208	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2386	. 2 |- F/_ x C
3	1, 2	nffv 5685	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2373   |-> cmpt 4176   ` cfv 5357

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 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-iota 5317  df-fv 5365

This theorem is referenced by:  fvmptt  5774  fmptco  5848  offval2  6291  ofrfval2  6292  mptelixpg  6982  dom2lem  7024  cc2  7597  fsumf1o  12101  fsum3cvg2  12105  fsumadd  12117  isummulc2  12137  isumshft  12201  fprodf1o  12299  prdsbas3  14129  txcnp  15262  cnmpt1t  15276  elplyd  15732