Theorem f1orescnv 3710

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 3707	. . 3 |- ((F |` R):R-1-1-onto->P -> `'(F |` R):P-1-1-onto->R)
2	1	adantl 390	. 2 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R):P-1-1-onto->R)
3		funcnvres 3574	. . . 4 |- (Fun `'F -> `'(F |` R) = (`'F |` (F"R)))
4		f1o5 3703	. . . . . . 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 3197	. . . . . 6 |- (F"R) = ran ( F |` R)
7	5, 6	syl5eq 1522	. . . . 5 |- ((F |` R):R-1-1-onto->P -> (F"R) = P)
8		reseq2 3375	. . . . 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 1530	. . 3 |- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> `'(F |` R) = (`'F |` P))
11		f1oeq1 3690	. . 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 958  `'ccnv 3175  ran crn 3177   |` cres 3178  "cima 3179  Fun wfun 3182  -1-1->wf1 3185  -1-1-onto->wf1o 3187

This theorem is referenced by:  relogf1o 8752

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-3an 779  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  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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