Theorem nffvmpt1 5701

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 4219	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2392	. 2 |- F/_ x C
3	1, 2	nffv 5700	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2379   |-> cmpt 4187   ` cfv 5372

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 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 theorem 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 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-iota 5332  df-fv 5380

This theorem is referenced by:  fvmptt  5791  fmptco  5865  offval2  6308  ofrfval2  6309  mptelixpg  7006  dom2lem  7048  cc2  7623  fsumf1o  12135  fsum3cvg2  12139  fsumadd  12151  isummulc2  12171  isumshft  12235  fprodf1o  12333  prdsbas3  14164  txcnp  15295  cnmpt1t  15309  elplyd  15765