Theorem nffvmpt1 5587

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 4137	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2348	. 2 |- F/_ x C
3	1, 2	nffv 5586	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2335   |-> cmpt 4105   ` cfv 5271

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 4045  df-opab 4106  df-mpt 4107  df-iota 5232  df-fv 5279

This theorem is referenced by:  fvmptt  5671  fmptco  5746  offval2  6174  ofrfval2  6175  mptelixpg  6821  dom2lem  6863  cc2  7379  fsumf1o  11701  fsum3cvg2  11705  fsumadd  11717  isummulc2  11737  isumshft  11801  fprodf1o  11899  prdsbas3  13119  txcnp  14743  cnmpt1t  14757  elplyd  15213