Theorem f1orescnv 3780

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 3777	. . 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 3643	. . . 4 |- (Fun `'F -> `'(F |` R) = (`'F |` (F"R)))
4		dff1o5 3773	. . . . . . 7 |- ((F |` R):R-1-1-onto->P <-> ((F |` R):R-1-1->P /\ ran ( F |` R) = P))
5	4	pm3.27bi 324	. . . . . 6 |- ((F |` R):R-1-1-onto->P -> ran ( F |` R) = P)
6		df-ima 3246	. . . . . 6 |- (F"R) = ran ( F |` R)
7	5, 6	syl5eq 1556	. . . . 5 |- ((F |` R):R-1-1-onto->P -> (F"R) = P)
8		reseq2 3429	. . . . 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 1564	. . 3 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R) = (`'F |` P))
11		f1oeq1 3760	. . 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 193	1 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> (`'F |` P):P-1-1-onto->R)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988  `'ccnv 3224  ran crn 3226   |` cres 3227  "cima 3228  Fun wfun 3231  -1-1->wf1 3234  -1-1-onto->wf1o 3236

This theorem is referenced by:  relogf1o 8876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252

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