Theorem f1orescnv 4655

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 4651	. . 3 |- ((F |` R):R-1-1-onto->P -> `'(F |` R):P-1-1-onto->R)
2	1	adantl 424	. 2 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R):P-1-1-onto->R)
3		funcnvres 4487	. . . 4 |- (Fun `'F -> `'(F |` R) = (`'F |` (F"R)))
4		dff1o5 4646	. . . . . . 7 |- ((F |` R):R-1-1-onto->P <-> ((F |` R):R-1-1->P /\ ran ( F |` R) = P))
5	4	simprbi 353	. . . . . 6 |- ((F |` R):R-1-1-onto->P -> ran ( F |` R) = P)
6		df-ima 4007	. . . . . 6 |- (F"R) = ran ( F |` R)
7	5, 6	syl5eq 1940	. . . . 5 |- ((F |` R):R-1-1-onto->P -> (F"R) = P)
8		reseq2 4219	. . . . 5 |- ((F"R) = P -> (`'F |` (F"R)) = (`'F |` P))
9	7, 8	syl 12	. . . 4 |- ((F |` R):R-1-1-onto->P -> (`'F |` (F"R)) = (`'F |` P))
10	3, 9	sylan9eq 1948	. . 3 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R) = (`'F |` P))
11		f1oeq1 4630	. . 3 |- (`'(F |` R) = (`'F |` P) -> (`'(F |` R):P-1-1-onto->R <-> (`'F |` P):P-1-1-onto->R))
12	10, 11	syl 12	. 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 212	1 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> (`'F |` P):P-1-1-onto->R)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  `'ccnv 3985  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992  -1-1->wf1 3995  -1-1-onto->wf1o 3997

This theorem is referenced by:  relogf1o 10111

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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