Theorem nffvmpt1 5638

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 4177	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2372	. 2 |- F/_ x C
3	1, 2	nffv 5637	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2359   |-> cmpt 4145   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-iota 5278  df-fv 5326

This theorem is referenced by:  fvmptt  5726  fmptco  5801  offval2  6234  ofrfval2  6235  mptelixpg  6881  dom2lem  6923  cc2  7453  fsumf1o  11901  fsum3cvg2  11905  fsumadd  11917  isummulc2  11937  isumshft  12001  fprodf1o  12099  prdsbas3  13320  txcnp  14945  cnmpt1t  14959  elplyd  15415