Theorem prl2 14514

Description: Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6.

Hypothesis

Ref	Expression
prl2.1	|- A e. D

Assertion

Ref	Expression
prl2	|- ((A.x e. A C =/= (/) /\ B C_ A) -> A.g e. X_ x e. B CE.f e. X_ x e. A Cg = (f |` B))

Distinct variable groups:   A,f,g   B,f,g   C,f,g   x,f,g,A   x,B

Proof of Theorem prl2

Step	Hyp	Ref	Expression
1		prl2.1	. . . 4 |- A e. D
2	1	prl1 14513	. . 3 |- ((A.x e. A C =/= (/) /\ B C_ A) -> A.g e. X_ x e. B CE.f e. X_ x e. A Cg C_ f)
3		elixp2a 14493	. . . . 5 |- (g e. X_x e. B C -> g Fn B)
4		fndm 4512	. . . . . 6 |- (g Fn B -> dom g = B)
5		elixp2a 14493	. . . . . . . . 9 |- (f e. X_x e. A C -> f Fn A)
6		fnfun 4510	. . . . . . . . 9 |- (f Fn A -> Fun f)
7		fvres 4691	. . . . . . . . . . . . . . 15 |- (u e. B -> ((f |` B)` u) = (f` u))
8		eqeq2 1893	. . . . . . . . . . . . . . . . 17 |- (((f |` B)` u) = (f` u) -> ((g` u) = ((f |` B)` u) <-> (g` u) = (f` u)))
9	8	imbi2d 674	. . . . . . . . . . . . . . . 16 |- (((f |` B)` u) = (f` u) -> (((Fun f /\ dom g = B /\ g C_ f) -> (g` u) = ((f |` B)` u)) <-> ((Fun f /\ dom g = B /\ g C_ f) -> (g` u) = (f` u))))
10		eleq2 1958	. . . . . . . . . . . . . . . . . . . . . 22 |- (B = dom g -> (u e. B <-> u e. dom g))
11		funssfv 4692	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((Fun f /\ g C_ f /\ u e. dom g) -> (f` u) = (g` u))
12	11	eqcomd 1889	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((Fun f /\ g C_ f /\ u e. dom g) -> (g` u) = (f` u))
13	12	3exp 1066	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Fun f -> (g C_ f -> (u e. dom g -> (g` u) = (f` u))))
14	13	com13 37	. . . . . . . . . . . . . . . . . . . . . 22 |- (u e. dom g -> (g C_ f -> (Fun f -> (g` u) = (f` u))))
15	10, 14	syl6bi 231	. . . . . . . . . . . . . . . . . . . . 21 |- (B = dom g -> (u e. B -> (g C_ f -> (Fun f -> (g` u) = (f` u)))))
16	15	eqcoms 1887	. . . . . . . . . . . . . . . . . . . 20 |- (dom g = B -> (u e. B -> (g C_ f -> (Fun f -> (g` u) = (f` u)))))
17	16	com24 41	. . . . . . . . . . . . . . . . . . 19 |- (dom g = B -> (Fun f -> (g C_ f -> (u e. B -> (g` u) = (f` u)))))
18	17	com12 14	. . . . . . . . . . . . . . . . . 18 |- (Fun f -> (dom g = B -> (g C_ f -> (u e. B -> (g` u) = (f` u)))))
19	18	3imp 1061	. . . . . . . . . . . . . . . . 17 |- ((Fun f /\ dom g = B /\ g C_ f) -> (u e. B -> (g` u) = (f` u)))
20	19	com12 14	. . . . . . . . . . . . . . . 16 |- (u e. B -> ((Fun f /\ dom g = B /\ g C_ f) -> (g` u) = (f` u)))
21	9, 20	syl5bir 227	. . . . . . . . . . . . . . 15 |- (((f |` B)` u) = (f` u) -> (u e. B -> ((Fun f /\ dom g = B /\ g C_ f) -> (g` u) = ((f |` B)` u))))
22	7, 21	mpcom 60	. . . . . . . . . . . . . 14 |- (u e. B -> ((Fun f /\ dom g = B /\ g C_ f) -> (g` u) = ((f |` B)` u)))
23	22	com12 14	. . . . . . . . . . . . 13 |- ((Fun f /\ dom g = B /\ g C_ f) -> (u e. B -> (g` u) = ((f |` B)` u)))
24	23	ancld 322	. . . . . . . . . . . 12 |- ((Fun f /\ dom g = B /\ g C_ f) -> (u e. B -> (u e. B /\ (g` u) = ((f |` B)` u))))
25	24	r19.21aiv 2175	. . . . . . . . . . 11 |- ((Fun f /\ dom g = B /\ g C_ f) -> A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))
26		ssid 2634	. . . . . . . . . . 11 |- B C_ B
27	25, 26	jctil 316	. . . . . . . . . 10 |- ((Fun f /\ dom g = B /\ g C_ f) -> (B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u))))
28	27	3exp 1066	. . . . . . . . 9 |- (Fun f -> (dom g = B -> (g C_ f -> (B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u))))))
29	5, 6, 28	3syl 24	. . . . . . . 8 |- (f e. X_x e. A C -> (dom g = B -> (g C_ f -> (B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u))))))
30	29	com12 14	. . . . . . 7 |- (dom g = B -> (f e. X_x e. A C -> (g C_ f -> (B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u))))))
31	30	reximdvai 2201	. . . . . 6 |- (dom g = B -> (E.f e. X_ x e. A Cg C_ f -> E.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
32	4, 31	syl 12	. . . . 5 |- (g Fn B -> (E.f e. X_ x e. A Cg C_ f -> E.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
33	3, 32	syl 12	. . . 4 |- (g e. X_x e. B C -> (E.f e. X_ x e. A Cg C_ f -> E.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
34	33	ralimia 2166	. . 3 |- (A.g e. X_ x e. B CE.f e. X_ x e. A Cg C_ f -> A.g e. X_ x e. B CE.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u))))
35	2, 34	syl 12	. 2 |- ((A.x e. A C =/= (/) /\ B C_ A) -> A.g e. X_ x e. B CE.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u))))
36	3	ad2antlr 441	. . . . 5 |- ((((A.x e. A C =/= (/) /\ B C_ A) /\ g e. X_x e. B C) /\ f e. X_x e. A C) -> g Fn B)
37		fnssresb 4525	. . . . . . . . 9 |- (f Fn A -> ((f |` B) Fn B <-> B C_ A))
38	37	biimprcd 173	. . . . . . . 8 |- (B C_ A -> (f Fn A -> (f |` B) Fn B))
39	38	ad2antlr 441	. . . . . . 7 |- (((A.x e. A C =/= (/) /\ B C_ A) /\ g e. X_x e. B C) -> (f Fn A -> (f |` B) Fn B))
40	39, 5	syl5 20	. . . . . 6 |- (((A.x e. A C =/= (/) /\ B C_ A) /\ g e. X_x e. B C) -> (f e. X_x e. A C -> (f |` B) Fn B))
41	40	imp 377	. . . . 5 |- ((((A.x e. A C =/= (/) /\ B C_ A) /\ g e. X_x e. B C) /\ f e. X_x e. A C) -> (f |` B) Fn B)
42		eqfnfv3 4769	. . . . 5 |- ((g Fn B /\ (f |` B) Fn B) -> (g = (f |` B) <-> (B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
43	36, 41, 42	syl11anc 524	. . . 4 |- ((((A.x e. A C =/= (/) /\ B C_ A) /\ g e. X_x e. B C) /\ f e. X_x e. A C) -> (g = (f |` B) <-> (B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
44	43	rexbidva 2120	. . 3 |- (((A.x e. A C =/= (/) /\ B C_ A) /\ g e. X_x e. B C) -> (E.f e. X_ x e. A Cg = (f |` B) <-> E.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
45	44	ralbidva 2119	. 2 |- ((A.x e. A C =/= (/) /\ B C_ A) -> (A.g e. X_ x e. B CE.f e. X_ x e. A Cg = (f |` B) <-> A.g e. X_ x e. B CE.f e. X_ x e. A C(B C_ B /\ A.u e. B (u e. B /\ (g` u) = ((f |` B)` u)))))
46	35, 45	mpbird 213	1 |- ((A.x e. A C =/= (/) /\ B C_ A) -> A.g e. X_ x e. B CE.f e. X_ x e. A Cg = (f |` B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  X_cixp 5406

This theorem is referenced by:  prjmapcp2 14515

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-fv 4014  df-rdg 5140  df-ixp 5407  df-r1 5750  df-rank 5751

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