Theorem prjmapcp2 14515

Description: A projection is a mapping from a cartesian product onto one of its restriction. Bourbaki E.II.33 prop. 5.

Assertion

Ref	Expression
prjmapcp2	|- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> (X_x e. A B prj I):X_x e. A B-onto->X_x e. I B)

Distinct variable groups:   x,A   x,I

Proof of Theorem prjmapcp2

Step	Hyp	Ref	Expression
1		dffo3 4792	. 2 |- ((X_x e. A B prj I):X_x e. A B-onto->X_x e. I B <-> ((X_x e. A B prj I):X_x e. A B-->X_x e. I B /\ A.y e. X_ x e. I BE.f e. X_ x e. A By = ((X_x e. A B prj I)` f)))
2		prjmapcp 14507	. . 3 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> (X_x e. A B prj I):X_x e. A B-->X_x e. I B)
3	2	3adant3r 1095	. 2 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> (X_x e. A B prj I):X_x e. A B-->X_x e. I B)
4		sseq2 2639	. . . . . . 7 |- (a = A -> (I C_ a <-> I C_ A))
5		raleq 2266	. . . . . . . . 9 |- (a = A -> (A.x e. a B e. D <-> A.x e. A B e. D))
6		raleq 2266	. . . . . . . . 9 |- (a = A -> (A.x e. a B =/= (/) <-> A.x e. A B =/= (/)))
7	5, 6	anbi12d 690	. . . . . . . 8 |- (a = A -> ((A.x e. a B e. D /\ A.x e. a B =/= (/)) <-> (A.x e. A B e. D /\ A.x e. A B =/= (/))))
8		ixpeq1 5412	. . . . . . . . . . . 12 |- (a = A -> X_x e. a B = X_x e. A B)
9	8	eleq2d 1964	. . . . . . . . . . 11 |- (a = A -> (f e. X_x e. a B <-> f e. X_x e. A B))
10	9	anbi1d 679	. . . . . . . . . 10 |- (a = A -> ((f e. X_x e. a B /\ y = (f |` I)) <-> (f e. X_x e. A B /\ y = (f |` I))))
11	10	rexbidv2 2126	. . . . . . . . 9 |- (a = A -> (E.f e. X_ x e. a By = (f |` I) <-> E.f e. X_ x e. A By = (f |` I)))
12	11	ralbidv 2123	. . . . . . . 8 |- (a = A -> (A.y e. X_ x e. I BE.f e. X_ x e. a By = (f |` I) <-> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I)))
13	7, 12	imbi12d 688	. . . . . . 7 |- (a = A -> (((A.x e. a B e. D /\ A.x e. a B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. a By = (f |` I)) <-> ((A.x e. A B e. D /\ A.x e. A B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I))))
14	4, 13	imbi12d 688	. . . . . 6 |- (a = A -> ((I C_ a -> ((A.x e. a B e. D /\ A.x e. a B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. a By = (f |` I))) <-> (I C_ A -> ((A.x e. A B e. D /\ A.x e. A B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I)))))
15		pm3.2 305	. . . . . . . 8 |- (A.x e. a B =/= (/) -> (I C_ a -> (A.x e. a B =/= (/) /\ I C_ a)))
16	15	adantl 424	. . . . . . 7 |- ((A.x e. a B e. D /\ A.x e. a B =/= (/)) -> (I C_ a -> (A.x e. a B =/= (/) /\ I C_ a)))
17		visset 2295	. . . . . . . 8 |- a e. _V
18	17	prl2 14514	. . . . . . 7 |- ((A.x e. a B =/= (/) /\ I C_ a) -> A.y e. X_ x e. I BE.f e. X_ x e. a By = (f |` I))
19	16, 18	syl6com 64	. . . . . 6 |- (I C_ a -> ((A.x e. a B e. D /\ A.x e. a B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. a By = (f |` I)))
20	14, 19	vtoclg 2346	. . . . 5 |- (A e. C -> (I C_ A -> ((A.x e. A B e. D /\ A.x e. A B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I))))
21	20	com12 14	. . . 4 |- (I C_ A -> (A e. C -> ((A.x e. A B e. D /\ A.x e. A B =/= (/)) -> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I))))
22	21	3imp 1061	. . 3 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I))
23		simpr 350	. . . . . . . . 9 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> f e. X_x e. A B)
24		visset 2295	. . . . . . . . . 10 |- f e. _V
25		resexg 4250	. . . . . . . . . 10 |- (f e. _V -> (f |` I) e. _V)
26	24, 25	ax-mp 7	. . . . . . . . 9 |- (f |` I) e. _V
27	23, 26	jctir 317	. . . . . . . 8 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> (f e. X_x e. A B /\ (f |` I) e. _V))
28		reseq1 4218	. . . . . . . . 9 |- (u = f -> (u |` I) = (f |` I))
29		eqid 1884	. . . . . . . . 9 |- {<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))} = {<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))}
30	28, 29	fvopab4g 4742	. . . . . . . 8 |- ((f e. X_x e. A B /\ (f |` I) e. _V) -> ({<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))}` f) = (f |` I))
31		simp2 877	. . . . . . . . . . . . . . . 16 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> A e. C)
32		simp3l 904	. . . . . . . . . . . . . . . 16 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> A.x e. A B e. D)
33	31, 32	jca 310	. . . . . . . . . . . . . . 15 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> (A e. C /\ A.x e. A B e. D))
34	33	adantr 425	. . . . . . . . . . . . . 14 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> (A e. C /\ A.x e. A B e. D))
35		ixpexg 5417	. . . . . . . . . . . . . 14 |- ((A e. C /\ A.x e. A B e. D) -> X_x e. A B e. _V)
36	34, 35	syl 12	. . . . . . . . . . . . 13 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> X_x e. A B e. _V)
37		ssexg 3457	. . . . . . . . . . . . . . 15 |- ((I C_ A /\ A e. C) -> I e. _V)
38	37	3adant3 896	. . . . . . . . . . . . . 14 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> I e. _V)
39	38	adantr 425	. . . . . . . . . . . . 13 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> I e. _V)
40		isprj1 14505	. . . . . . . . . . . . 13 |- ((X_x e. A B e. _V /\ I e. _V) -> (X_x e. A B prj I) = {<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))})
41	36, 39, 40	syl11anc 524	. . . . . . . . . . . 12 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> (X_x e. A B prj I) = {<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))})
42	41	eqcomd 1889	. . . . . . . . . . 11 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> {<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))} = (X_x e. A B prj I))
43	42	fveq1d 4683	. . . . . . . . . 10 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> ({<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))}` f) = ((X_x e. A B prj I)` f))
44	43	eqeq1d 1892	. . . . . . . . 9 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> (({<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))}` f) = (f |` I) <-> ((X_x e. A B prj I)` f) = (f |` I)))
45	44	biimpcd 172	. . . . . . . 8 |- (({<.u, v>. | (u e. X_x e. A B /\ v = (u |` I))}` f) = (f |` I) -> (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> ((X_x e. A B prj I)` f) = (f |` I)))
46	27, 30, 45	3syl 24	. . . . . . 7 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> ((X_x e. A B prj I)` f) = (f |` I)))
47	46	pm2.43i 78	. . . . . 6 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> ((X_x e. A B prj I)` f) = (f |` I))
48	47	eqeq2d 1895	. . . . 5 |- (((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) /\ f e. X_x e. A B) -> (y = ((X_x e. A B prj I)` f) <-> y = (f |` I)))
49	48	rexbidva 2120	. . . 4 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> (E.f e. X_ x e. A By = ((X_x e. A B prj I)` f) <-> E.f e. X_ x e. A By = (f |` I)))
50	49	ralbidv 2123	. . 3 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> (A.y e. X_ x e. I BE.f e. X_ x e. A By = ((X_x e. A B prj I)` f) <-> A.y e. X_ x e. I BE.f e. X_ x e. A By = (f |` I)))
51	22, 50	mpbird 213	. 2 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> A.y e. X_ x e. I BE.f e. X_ x e. A By = ((X_x e. A B prj I)` f))
52	1, 3, 51	sylanbrc 527	1 |- ((I C_ A /\ A e. C /\ (A.x e. A B e. D /\ A.x e. A B =/= (/))) -> (X_x e. A B prj I):X_x e. A B-onto->X_x e. I B)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {copab 3395   |` cres 3988  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  X_cixp 5406   prj cproj 14490

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-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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  df-opr 4886  df-oprab 4887  df-rdg 5140  df-ixp 5407  df-r1 5750  df-rank 5751  df-prj 14492

Copyright terms: Public domain