Theorem injrec2 14466

Description: A function is an injection iff a retraction exists. Bourbaki E.II.18 prop. 8.

Assertion

Ref	Expression
injrec2	|- ((F:A-->B /\ A e. C) -> (F:A-1-1->B <-> E.r(Fun r /\ (r o. F) = ( _I |` A))))

Distinct variable groups:   A,r   B,r   F,r

Proof of Theorem injrec2

Step	Hyp	Ref	Expression
1		fex 4595	. . . . . . 7 |- ((F:A-->B /\ A e. C) -> F e. _V)
2		cnvexg 4424	. . . . . . 7 |- (F e. _V -> `'F e. _V)
3	1, 2	syl 12	. . . . . 6 |- ((F:A-->B /\ A e. C) -> `'F e. _V)
4	3	adantr 425	. . . . 5 |- (((F:A-->B /\ A e. C) /\ F:A-1-1->B) -> `'F e. _V)
5		df-f1 4011	. . . . . . 7 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
6	5	simprbi 353	. . . . . 6 |- (F:A-1-1->B -> Fun `'F)
7	6	adantl 424	. . . . 5 |- (((F:A-->B /\ A e. C) /\ F:A-1-1->B) -> Fun `'F)
8		cmpinj 14464	. . . . . 6 |- (F:A-1-1->B -> (`'F o. F) = ( _I |` A))
9	8	adantl 424	. . . . 5 |- (((F:A-->B /\ A e. C) /\ F:A-1-1->B) -> (`'F o. F) = ( _I |` A))
10	4, 7, 9	jca32 312	. . . 4 |- (((F:A-->B /\ A e. C) /\ F:A-1-1->B) -> (`'F e. _V /\ (Fun `'F /\ (`'F o. F) = ( _I |` A))))
11	10	ex 402	. . 3 |- ((F:A-->B /\ A e. C) -> (F:A-1-1->B -> (`'F e. _V /\ (Fun `'F /\ (`'F o. F) = ( _I |` A)))))
12		funeq 4441	. . . . . 6 |- (r = `'F -> (Fun r <-> Fun `'F))
13		coeq1 4123	. . . . . . 7 |- (r = `'F -> (r o. F) = (`'F o. F))
14	13	eqeq1d 1892	. . . . . 6 |- (r = `'F -> ((r o. F) = ( _I |` A) <-> (`'F o. F) = ( _I |` A)))
15	12, 14	anbi12d 690	. . . . 5 |- (r = `'F -> ((Fun r /\ (r o. F) = ( _I |` A)) <-> (Fun `'F /\ (`'F o. F) = ( _I |` A))))
16	15	cla4egv 2365	. . . 4 |- (`'F e. _V -> ((Fun `'F /\ (`'F o. F) = ( _I |` A)) -> E.r(Fun r /\ (r o. F) = ( _I |` A))))
17	16	imp 377	. . 3 |- ((`'F e. _V /\ (Fun `'F /\ (`'F o. F) = ( _I |` A))) -> E.r(Fun r /\ (r o. F) = ( _I |` A)))
18	11, 17	syl6 25	. 2 |- ((F:A-->B /\ A e. C) -> (F:A-1-1->B -> E.r(Fun r /\ (r o. F) = ( _I |` A))))
19		injrec 14461	. . . . . . . 8 |- ((F:A-->B /\ Fun r /\ (r o. F) = ( _I |` A)) -> F:A-1-1->B)
20	19	3exp 1066	. . . . . . 7 |- (F:A-->B -> (Fun r -> ((r o. F) = ( _I |` A) -> F:A-1-1->B)))
21	20	com3l 38	. . . . . 6 |- (Fun r -> ((r o. F) = ( _I |` A) -> (F:A-->B -> F:A-1-1->B)))
22	21	imp 377	. . . . 5 |- ((Fun r /\ (r o. F) = ( _I |` A)) -> (F:A-->B -> F:A-1-1->B))
23	22	19.23aiv 1674	. . . 4 |- (E.r(Fun r /\ (r o. F) = ( _I |` A)) -> (F:A-->B -> F:A-1-1->B))
24	23	com12 14	. . 3 |- (F:A-->B -> (E.r(Fun r /\ (r o. F) = ( _I |` A)) -> F:A-1-1->B))
25	24	adantr 425	. 2 |- ((F:A-->B /\ A e. C) -> (E.r(Fun r /\ (r o. F) = ( _I |` A)) -> F:A-1-1->B))
26	18, 25	impbid 574	1 |- ((F:A-->B /\ A e. C) -> (F:A-1-1->B <-> E.r(Fun r /\ (r o. F) = ( _I |` A))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   _I cid 3582  `'ccnv 3985   |` cres 3988   o. ccom 3990  Fun wfun 3992  -->wf 3994  -1-1->wf1 3995

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

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-ral 2109  df-rex 2110  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-uni 3178  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  df-fv 4014

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