Theorem funcnvres2 5273

Description: The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)

Assertion

Ref	Expression
funcnvres2	|- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )

Proof of Theorem funcnvres2

Step	Hyp	Ref	Expression
1		funcnvcnv 5257	. . 3 |- ( Fun F -> Fun `' `' F )
2		funcnvres 5271	. . 3 |- ( Fun `' `' F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
3	1, 2	syl 14	. 2 |- ( Fun F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
4		funrel 5215	. . . 4 |- ( Fun F -> Rel F )
5		dfrel2 5061	. . . 4 |- ( Rel F <-> `' `' F = F )
6	4, 5	sylib 121	. . 3 |- ( Fun F -> `' `' F = F )
7	6	reseq1d 4890	. 2 |- ( Fun F -> ( `' `' F |` ( `' F " A ) ) = ( F |` ( `' F " A ) ) )
8	3, 7	eqtrd 2203	1 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1348   `'ccnv 4610   |` cres 4613   "cima 4614   Rel wrel 4616   Fun wfun 5192

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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200

This theorem is referenced by:  funimacnv  5274  foimacnv  5460