Theorem resresdm 5626

Description: A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)

Assertion

Ref	Expression
resresdm	|- ( F = ( E |` A ) -> F = ( E |` dom F ) )

Proof of Theorem resresdm

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( F = ( E |` A ) -> F = ( E |` A ) )
2		dmeq 5324	. . . 4 |- ( F = ( E |` A ) -> dom F = dom ( E |` A ) )
3	2	reseq2d 5396	. . 3 |- ( F = ( E |` A ) -> ( E |` dom F ) = ( E |` dom ( E |` A ) ) )
4		resdmres 5625	. . 3 |- ( E |` dom ( E |` A ) ) = ( E |` A )
5	3, 4	syl6req 2673	. 2 |- ( F = ( E |` A ) -> ( E |` A ) = ( E |` dom F ) )
6	1, 5	eqtrd 2656	1 |- ( F = ( E |` A ) -> F = ( E |` dom F ) )

Syntax hints:   -> wi 4   = wceq 1483   dom cdm 5114   |` cres 5116

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-eu 2474  df-mo 2475  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  uhgrspan1  26195