Theorem funcnvres2 3576

Description: The converse of a restriction of the converse of a function equals the function restricted to the image of its converse.

Assertion

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

Proof of Theorem funcnvres2

Step	Hyp	Ref	Expression
1		funcnvcnv 3561	. . 3 |- (Fun F -> Fun `'`'F)
2		funcnvres 3574	. . 3 |- (Fun `'`'F -> `'(`'F |` A) = (`'`'F |` (`'F"A)))
3	1, 2	syl 10	. 2 |- (Fun F -> `'(`'F |` A) = (`'`'F |` (`'F"A)))
4		funrel 3539	. . . 4 |- (Fun F -> Rel F)
5		dfrel2 3491	. . . 4 |- (Rel F <-> `'`'F = F)
6	4, 5	sylib 198	. . 3 |- (Fun F -> `'`'F = F)
7		reseq1 3374	. . 3 |- (`'`'F = F -> (`'`'F |` (`'F"A)) = (F |` (`'F"A)))
8	6, 7	syl 10	. 2 |- (Fun F -> (`'`'F |` (`'F"A)) = (F |` (`'F"A)))
9	3, 8	eqtrd 1510	1 |- (Fun F -> `'(`'F |` A) = (F |` (`'F"A)))

Syntax hints:   -> wi 3   = wceq 958  `'ccnv 3175   |` cres 3178  "cima 3179  Rel wrel 3181  Fun wfun 3182

This theorem is referenced by:  funimacnv 3577  unbenlem 7505

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198

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