Theorem f1orescnv 3689

Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Assertion

Ref	Expression
f1orescnv	|- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> (`'F |` P):P-1-1-onto->R)

Proof of Theorem f1orescnv

Step	Hyp	Ref	Expression
1		f1ocnv 3686	. . 3 |- ((F |` R):R-1-1-onto->P -> `'(F |` R):P-1-1-onto->R)
2	1	adantl 388	. 2 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R):P-1-1-onto->R)
3		funcnvres 3554	. . . 4 |- (Fun `'F -> `'(F |` R) = (`'F |` (F"R)))
4		f1o5 3682	. . . . . . 7 |- ((F |` R):R-1-1-onto->P <-> ((F |` R):R-1-1->P /\ ran ( F |` R) = P))
5	4	pm3.27bi 326	. . . . . 6 |- ((F |` R):R-1-1-onto->P -> ran ( F |` R) = P)
6		df-ima 3181	. . . . . 6 |- (F"R) = ran ( F |` R)
7	5, 6	syl5eq 1511	. . . . 5 |- ((F |` R):R-1-1-onto->P -> (F"R) = P)
8		reseq2 3353	. . . . 5 |- ((F"R) = P -> (`'F |` (F"R)) = (`'F |` P))
9	7, 8	syl 10	. . . 4 |- ((F |` R):R-1-1-onto->P -> (`'F |` (F"R)) = (`'F |` P))
10	3, 9	sylan9eq 1519	. . 3 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R) = (`'F |` P))
11		f1oeq1 3669	. . 3 |- (`'(F |` R) = (`'F |` P) -> (`'(F |` R):P-1-1-onto->R <-> (`'F |` P):P-1-1-onto->R))
12	10, 11	syl 10	. 2 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> (`'(F |` R):P-1-1-onto->R <-> (`'F |` P):P-1-1-onto->R))
13	2, 12	mpbid 195	1 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> (`'F |` P):P-1-1-onto->R)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953  `'ccnv 3159  ran crn 3161   |` cres 3162  "cima 3163  Fun wfun 3166  -1-1->wf1 3169  -1-1-onto->wf1o 3171

This theorem is referenced by:  relogf1o 8679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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