Theorem funcnvres2 3570

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 3555	. . 3 |- (Fun F -> Fun `'`'F)
2		funcnvres 3568	. . 3 |- (Fun `'`'F -> `'(`'F |` A) = (`'`'F |` (`'F"A)))
3	1, 2	syl 10	. 2 |- (Fun F -> `'(`'F |` A) = (`'`'F |` (`'F"A)))
4		funrel 3533	. . . 4 |- (Fun F -> Rel F)
5		dfrel2 3485	. . . 4 |- (Rel F <-> `'`'F = F)
6	4, 5	sylib 198	. . 3 |- (Fun F -> `'`'F = F)
7		reseq1 3368	. . 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 1507	1 |- (Fun F -> `'(`'F |` A) = (F |` (`'F"A)))

Syntax hints:   -> wi 3   = wceq 956  `'ccnv 3169   |` cres 3172  "cima 3173  Rel wrel 3175  Fun wfun 3176

This theorem is referenced by:  funimacnv 3571  unbenlem 7504

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192

Copyright terms: Public domain