Theorem surjsec2 14467

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

Assertion

Ref	Expression
surjsec2	|- ((F:A-->B /\ A e. C /\ B e. D) -> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B))))

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

Proof of Theorem surjsec2

Step	Hyp	Ref	Expression
1		feq2 4552	. . . . . 6 |- (a = A -> (F:a-->b <-> F:A-->b))
2		foeq2 4614	. . . . . . 7 |- (a = A -> (F:a-onto->b <-> F:A-onto->b))
3		feq3 4553	. . . . . . . . 9 |- (a = A -> (s:b-->a <-> s:b-->A))
4	3	anbi1d 679	. . . . . . . 8 |- (a = A -> ((s:b-->a /\ (F o. s) = ( _I |` b)) <-> (s:b-->A /\ (F o. s) = ( _I |` b))))
5	4	exbidv 1657	. . . . . . 7 |- (a = A -> (E.s(s:b-->a /\ (F o. s) = ( _I |` b)) <-> E.s(s:b-->A /\ (F o. s) = ( _I |` b))))
6	2, 5	bibi12d 691	. . . . . 6 |- (a = A -> ((F:a-onto->b <-> E.s(s:b-->a /\ (F o. s) = ( _I |` b))) <-> (F:A-onto->b <-> E.s(s:b-->A /\ (F o. s) = ( _I |` b)))))
7	1, 6	imbi12d 688	. . . . 5 |- (a = A -> ((F:a-->b -> (F:a-onto->b <-> E.s(s:b-->a /\ (F o. s) = ( _I |` b)))) <-> (F:A-->b -> (F:A-onto->b <-> E.s(s:b-->A /\ (F o. s) = ( _I |` b))))))
8		feq3 4553	. . . . . 6 |- (b = B -> (F:A-->b <-> F:A-->B))
9		foeq3 4615	. . . . . . 7 |- (b = B -> (F:A-onto->b <-> F:A-onto->B))
10		feq2 4552	. . . . . . . . 9 |- (b = B -> (s:b-->A <-> s:B-->A))
11		reseq2 4219	. . . . . . . . . 10 |- (b = B -> ( _I |` b) = ( _I |` B))
12	11	eqeq2d 1895	. . . . . . . . 9 |- (b = B -> ((F o. s) = ( _I |` b) <-> (F o. s) = ( _I |` B)))
13	10, 12	anbi12d 690	. . . . . . . 8 |- (b = B -> ((s:b-->A /\ (F o. s) = ( _I |` b)) <-> (s:B-->A /\ (F o. s) = ( _I |` B))))
14	13	exbidv 1657	. . . . . . 7 |- (b = B -> (E.s(s:b-->A /\ (F o. s) = ( _I |` b)) <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B))))
15	9, 14	bibi12d 691	. . . . . 6 |- (b = B -> ((F:A-onto->b <-> E.s(s:b-->A /\ (F o. s) = ( _I |` b))) <-> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B)))))
16	8, 15	imbi12d 688	. . . . 5 |- (b = B -> ((F:A-->b -> (F:A-onto->b <-> E.s(s:b-->A /\ (F o. s) = ( _I |` b)))) <-> (F:A-->B -> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B))))))
17		dffo3 4792	. . . . . . . . 9 |- (F:a-onto->b <-> (F:a-->b /\ A.x e. b E.y e. a x = (F` y)))
18		eqcom 1886	. . . . . . . . . . . . 13 |- (x = (F` y) <-> (F` y) = x)
19	18	biimpi 168	. . . . . . . . . . . 12 |- (x = (F` y) -> (F` y) = x)
20	19	reximi 2198	. . . . . . . . . . 11 |- (E.y e. a x = (F` y) -> E.y e. a (F` y) = x)
21	20	ralimi 2168	. . . . . . . . . 10 |- (A.x e. b E.y e. a x = (F` y) -> A.x e. b E.y e. a (F` y) = x)
22	21	adantl 424	. . . . . . . . 9 |- ((F:a-->b /\ A.x e. b E.y e. a x = (F` y)) -> A.x e. b E.y e. a (F` y) = x)
23	17, 22	sylbi 216	. . . . . . . 8 |- (F:a-onto->b -> A.x e. b E.y e. a (F` y) = x)
24		visset 2295	. . . . . . . . 9 |- b e. _V
25		visset 2295	. . . . . . . . 9 |- a e. _V
26		fveq2 4681	. . . . . . . . . 10 |- (y = (s` x) -> (F` y) = (F` (s` x)))
27	26	eqeq1d 1892	. . . . . . . . 9 |- (y = (s` x) -> ((F` y) = x <-> (F` (s` x)) = x))
28	24, 25, 27	ac6 5917	. . . . . . . 8 |- (A.x e. b E.y e. a (F` y) = x -> E.s(s:b-->a /\ A.x e. b (F` (s` x)) = x))
29	23, 28	syl 12	. . . . . . 7 |- (F:a-onto->b -> E.s(s:b-->a /\ A.x e. b (F` (s` x)) = x))
30		eqfnfv2 4767	. . . . . . . . . . 11 |- (((F o. s) Fn b /\ ( _I |` b) Fn b) -> ((F o. s) = ( _I |` b) <-> A.x e. b ((F o. s)` x) = (( _I |` b)` x)))
31		fnfco 4581	. . . . . . . . . . . 12 |- ((F Fn a /\ s:b-->a) -> (F o. s) Fn b)
32		fofn 4619	. . . . . . . . . . . 12 |- (F:a-onto->b -> F Fn a)
33	31, 32	sylan 497	. . . . . . . . . . 11 |- ((F:a-onto->b /\ s:b-->a) -> (F o. s) Fn b)
34		fnresi 4529	. . . . . . . . . . 11 |- ( _I |` b) Fn b
35	30, 33, 34	sylancl 525	. . . . . . . . . 10 |- ((F:a-onto->b /\ s:b-->a) -> ((F o. s) = ( _I |` b) <-> A.x e. b ((F o. s)` x) = (( _I |` b)` x)))
36		fofun 4618	. . . . . . . . . . . . . . 15 |- (F:a-onto->b -> Fun F)
37	36	3ad2ant1 897	. . . . . . . . . . . . . 14 |- ((F:a-onto->b /\ s:b-->a /\ x e. b) -> Fun F)
38		ffun 4565	. . . . . . . . . . . . . . 15 |- (s:b-->a -> Fun s)
39	38	3ad2ant2 898	. . . . . . . . . . . . . 14 |- ((F:a-onto->b /\ s:b-->a /\ x e. b) -> Fun s)
40		fdm 4567	. . . . . . . . . . . . . . . . . 18 |- (s:b-->a -> dom s = b)
41	40	eqcomd 1889	. . . . . . . . . . . . . . . . 17 |- (s:b-->a -> b = dom s)
42	41	eleq2d 1964	. . . . . . . . . . . . . . . 16 |- (s:b-->a -> (x e. b <-> x e. dom s))
43	42	biimpa 460	. . . . . . . . . . . . . . 15 |- ((s:b-->a /\ x e. b) -> x e. dom s)
44	43	3adant1 894	. . . . . . . . . . . . . 14 |- ((F:a-onto->b /\ s:b-->a /\ x e. b) -> x e. dom s)
45		fvco 4736	. . . . . . . . . . . . . 14 |- ((Fun F /\ Fun s /\ x e. dom s) -> ((F o. s)` x) = (F` (s` x)))
46	37, 39, 44, 45	syl111anc 1100	. . . . . . . . . . . . 13 |- ((F:a-onto->b /\ s:b-->a /\ x e. b) -> ((F o. s)` x) = (F` (s` x)))
47		fvresi 4819	. . . . . . . . . . . . . 14 |- (x e. b -> (( _I |` b)` x) = x)
48	47	3ad2ant3 899	. . . . . . . . . . . . 13 |- ((F:a-onto->b /\ s:b-->a /\ x e. b) -> (( _I |` b)` x) = x)
49	46, 48	eqeq12d 1899	. . . . . . . . . . . 12 |- ((F:a-onto->b /\ s:b-->a /\ x e. b) -> (((F o. s)` x) = (( _I |` b)` x) <-> (F` (s` x)) = x))
50	49	3expa 1067	. . . . . . . . . . 11 |- (((F:a-onto->b /\ s:b-->a) /\ x e. b) -> (((F o. s)` x) = (( _I |` b)` x) <-> (F` (s` x)) = x))
51	50	ralbidva 2119	. . . . . . . . . 10 |- ((F:a-onto->b /\ s:b-->a) -> (A.x e. b ((F o. s)` x) = (( _I |` b)` x) <-> A.x e. b (F` (s` x)) = x))
52	35, 51	bitrd 587	. . . . . . . . 9 |- ((F:a-onto->b /\ s:b-->a) -> ((F o. s) = ( _I |` b) <-> A.x e. b (F` (s` x)) = x))
53	52	pm5.32da 711	. . . . . . . 8 |- (F:a-onto->b -> ((s:b-->a /\ (F o. s) = ( _I |` b)) <-> (s:b-->a /\ A.x e. b (F` (s` x)) = x)))
54	53	exbidv 1657	. . . . . . 7 |- (F:a-onto->b -> (E.s(s:b-->a /\ (F o. s) = ( _I |` b)) <-> E.s(s:b-->a /\ A.x e. b (F` (s` x)) = x)))
55	29, 54	mpbird 213	. . . . . 6 |- (F:a-onto->b -> E.s(s:b-->a /\ (F o. s) = ( _I |` b)))
56		surjsec 14462	. . . . . . . . . . 11 |- ((F:a-->b /\ s:b-->a /\ (F o. s) = ( _I |` b)) -> F:a-onto->b)
57	56	3exp 1066	. . . . . . . . . 10 |- (F:a-->b -> (s:b-->a -> ((F o. s) = ( _I |` b) -> F:a-onto->b)))
58	57	com3l 38	. . . . . . . . 9 |- (s:b-->a -> ((F o. s) = ( _I |` b) -> (F:a-->b -> F:a-onto->b)))
59	58	imp 377	. . . . . . . 8 |- ((s:b-->a /\ (F o. s) = ( _I |` b)) -> (F:a-->b -> F:a-onto->b))
60	59	19.23aiv 1674	. . . . . . 7 |- (E.s(s:b-->a /\ (F o. s) = ( _I |` b)) -> (F:a-->b -> F:a-onto->b))
61	60	com12 14	. . . . . 6 |- (F:a-->b -> (E.s(s:b-->a /\ (F o. s) = ( _I |` b)) -> F:a-onto->b))
62	55, 61	impbid2 576	. . . . 5 |- (F:a-->b -> (F:a-onto->b <-> E.s(s:b-->a /\ (F o. s) = ( _I |` b))))
63	7, 16, 62	vtocl2g 2349	. . . 4 |- ((A e. C /\ B e. D) -> (F:A-->B -> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B)))))
64	63	ex 402	. . 3 |- (A e. C -> (B e. D -> (F:A-->B -> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B))))))
65	64	com3r 39	. 2 |- (F:A-->B -> (A e. C -> (B e. D -> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B))))))
66	65	3imp 1061	1 |- ((F:A-->B /\ A e. C /\ B e. D) -> (F:A-onto->B <-> E.s(s:B-->A /\ (F o. s) = ( _I |` B))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106   _I cid 3582  dom cdm 3986   |` cres 3988   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998

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  ax-ac 5906

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-reu 2111  df-rab 2112  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-fo 4012  df-fv 4014

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