Theorem fresin2 39352

Description: Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
fresin2	|- ( F : A --> B -> ( F |` ( C i^i A ) ) = ( F |` C ) )

Proof of Theorem fresin2

Step	Hyp	Ref	Expression
1		fdm 6051	. . . . 5 |- ( F : A --> B -> dom F = A )
2	1	eqcomd 2628	. . . 4 |- ( F : A --> B -> A = dom F )
3	2	ineq2d 3814	. . 3 |- ( F : A --> B -> ( C i^i A ) = ( C i^i dom F ) )
4	3	reseq2d 5396	. 2 |- ( F : A --> B -> ( F |` ( C i^i A ) ) = ( F |` ( C i^i dom F ) ) )
5		frel 6050	. . 3 |- ( F : A --> B -> Rel F )
6		resindm 5444	. . 3 |- ( Rel F -> ( F |` ( C i^i dom F ) ) = ( F |` C ) )
7	5, 6	syl 17	. 2 |- ( F : A --> B -> ( F |` ( C i^i dom F ) ) = ( F |` C ) )
8	4, 7	eqtrd 2656	1 |- ( F : A --> B -> ( F |` ( C i^i A ) ) = ( F |` C ) )

Syntax hints:   -> wi 4   = wceq 1483   i^i cin 3573   dom cdm 5114   |` cres 5116   Rel wrel 5119   -->wf 5884

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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-res 5126  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by: (None)