Theorem nffvmpt1 5589

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 4138	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2348	. 2 |- F/_ x C
3	1, 2	nffv 5588	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2335   |-> cmpt 4106   ` cfv 5272

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-iota 5233  df-fv 5280

This theorem is referenced by:  fvmptt  5673  fmptco  5748  offval2  6176  ofrfval2  6177  mptelixpg  6823  dom2lem  6865  cc2  7381  fsumf1o  11734  fsum3cvg2  11738  fsumadd  11750  isummulc2  11770  isumshft  11834  fprodf1o  11932  prdsbas3  13152  txcnp  14776  cnmpt1t  14790  elplyd  15246