Theorem fvresd 6208

Description: The value of a restricted function, deduction version of fvres 6207. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypothesis

Ref	Expression
fvresd.1	|- ( ph -> A e. B )

Assertion

Ref	Expression
fvresd	|- ( ph -> ( ( F |` B ) ` A ) = ( F ` A ) )

Proof of Theorem fvresd

Step	Hyp	Ref	Expression
1		fvresd.1	. 2 |- ( ph -> A e. B )
2		fvres 6207	. 2 |- ( A e. B -> ( ( F |` B ) ` A ) = ( F ` A ) )
3	1, 2	syl 17	1 |- ( ph -> ( ( F |` B ) ` A ) = ( F ` A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |` cres 5116   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  ackbij2lem2  9062  cfsmolem  9092  txkgen  21455  loglesqrt  24499  uhgrspansubgrlem  26182  wlkres  26567  ftc2re  30676  reprsuc  30693  nolesgn2o  31824  nolesgn2ores  31825  noresle  31846  noprefixmo  31848  nosupres  31853  nosupbnd2lem1  31861  noetalem3  31865  limsupresxr  39998  liminfresxr  39999  sssmf  40947