Theorem nffvmpt1 5706

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 4224	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2392	. 2 |- F/_ x C
3	1, 2	nffv 5705	1 |- F/_ x ( ( x e. A |-> B ) ` C )

This proof depends on syntax axioms:   F/_wnfc 2379   |-> cmpt 4192   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-iota 5337  df-fv 5385

This theorem is used by:  fvmptt  5797  fmptco  5874  offval2  6318  ofrfval2  6319  mptelixpg  7016  dom2lem  7058  cc2  7633  fsumf1o  12157  fsum3cvg2  12161  fsumadd  12173  isummulc2  12193  isumshft  12257  fprodf1o  12355  prdsbas3  14187  txcnp  15372  cnmpt1t  15386  elplyd  15842