Theorem nffvmpt1 5505

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 4080	. 2 |- F/_ x ( x e. A |-> B )
2		nfcv 2312	. 2 |- F/_ x C
3	1, 2	nffv 5504	1 |- F/_ x ( ( x e. A |-> B ) ` C )

Syntax hints:   F/_wnfc 2299   |-> cmpt 4048   ` cfv 5196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-iota 5158  df-fv 5204

This theorem is referenced by:  fvmptt  5585  fmptco  5660  offval2  6074  ofrfval2  6075  mptelixpg  6709  dom2lem  6747  cc2  7218  fsumf1o  11342  fsum3cvg2  11346  fsumadd  11358  isummulc2  11378  isumshft  11442  fprodf1o  11540  txcnp  13026  cnmpt1t  13040