Theorem prl 14512

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

Hypothesis

Ref	Expression
prl.1	|- A e. D

Assertion

Ref	Expression
prl	|- ((A.x e. A C =/= (/) /\ B C_ A /\ G e. X_x e. B C) -> E.f e. X_ x e. A CG C_ f)

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

Proof of Theorem prl

Step	Hyp	Ref	Expression
1		prl.1	. . . . . . 7 |- A e. D
2	1	elisseti 2301	. . . . . 6 |- A e. _V
3	2	ac9s 5926	. . . . 5 |- (A.x e. A C =/= (/) <-> X_x e. A C =/= (/))
4		n0 2884	. . . . . 6 |- (X_x e. A C =/= (/) <-> E.f f e. X_x e. A C)
5		difss 2735	. . . . . . . . 9 |- (A \ B) C_ A
6		prjnpl 14510	. . . . . . . . . 10 |- (((A \ B) C_ A /\ f e. X_x e. A C) -> (f |` (A \ B)) e. X_x e. (A \ B)C)
7		eleq1 1957	. . . . . . . . . . 11 |- (h = (f |` (A \ B)) -> (h e. X_x e. (A \ B)C <-> (f |` (A \ B)) e. X_x e. (A \ B)C))
8	7	cla4egv 2365	. . . . . . . . . 10 |- ((f |` (A \ B)) e. X_x e. (A \ B)C -> ((f |` (A \ B)) e. X_x e. (A \ B)C -> E.h h e. X_x e. (A \ B)C))
9	6, 6, 8	sylc 83	. . . . . . . . 9 |- (((A \ B) C_ A /\ f e. X_x e. A C) -> E.h h e. X_x e. (A \ B)C)
10	5, 9	mpan 759	. . . . . . . 8 |- (f e. X_x e. A C -> E.h h e. X_x e. (A \ B)C)
11	10	a1d 15	. . . . . . 7 |- (f e. X_x e. A C -> (B C_ A -> E.h h e. X_x e. (A \ B)C))
12	11	19.23aiv 1674	. . . . . 6 |- (E.f f e. X_x e. A C -> (B C_ A -> E.h h e. X_x e. (A \ B)C))
13	4, 12	sylbi 216	. . . . 5 |- (X_x e. A C =/= (/) -> (B C_ A -> E.h h e. X_x e. (A \ B)C))
14	3, 13	sylbi 216	. . . 4 |- (A.x e. A C =/= (/) -> (B C_ A -> E.h h e. X_x e. (A \ B)C))
15	14	imp 377	. . 3 |- ((A.x e. A C =/= (/) /\ B C_ A) -> E.h h e. X_x e. (A \ B)C)
16	15	3adant3 896	. 2 |- ((A.x e. A C =/= (/) /\ B C_ A /\ G e. X_x e. B C) -> E.h h e. X_x e. (A \ B)C)
17		unprj 14511	. . . . . . . . 9 |- ((G e. X_x e. B C /\ h e. X_x e. (A \ B)C /\ B C_ A) -> (G u. h) e. X_x e. A C)
18		ssun1 2767	. . . . . . . . . . 11 |- G C_ (G u. h)
19		sseq2 2639	. . . . . . . . . . . 12 |- (f = (G u. h) -> (G C_ f <-> G C_ (G u. h)))
20	19	rcla4ev 2381	. . . . . . . . . . 11 |- (((G u. h) e. X_x e. A C /\ G C_ (G u. h)) -> E.f e. X_ x e. A CG C_ f)
21	18, 20	mpan2 760	. . . . . . . . . 10 |- ((G u. h) e. X_x e. A C -> E.f e. X_ x e. A CG C_ f)
22	21	a1d 15	. . . . . . . . 9 |- ((G u. h) e. X_x e. A C -> (A.x e. A C =/= (/) -> E.f e. X_ x e. A CG C_ f))
23	17, 22	syl 12	. . . . . . . 8 |- ((G e. X_x e. B C /\ h e. X_x e. (A \ B)C /\ B C_ A) -> (A.x e. A C =/= (/) -> E.f e. X_ x e. A CG C_ f))
24	23	3exp 1066	. . . . . . 7 |- (G e. X_x e. B C -> (h e. X_x e. (A \ B)C -> (B C_ A -> (A.x e. A C =/= (/) -> E.f e. X_ x e. A CG C_ f))))
25	24	com34 40	. . . . . 6 |- (G e. X_x e. B C -> (h e. X_x e. (A \ B)C -> (A.x e. A C =/= (/) -> (B C_ A -> E.f e. X_ x e. A CG C_ f))))
26	25	com4t 44	. . . . 5 |- (A.x e. A C =/= (/) -> (B C_ A -> (G e. X_x e. B C -> (h e. X_x e. (A \ B)C -> E.f e. X_ x e. A CG C_ f))))
27	26	3imp 1061	. . . 4 |- ((A.x e. A C =/= (/) /\ B C_ A /\ G e. X_x e. B C) -> (h e. X_x e. (A \ B)C -> E.f e. X_ x e. A CG C_ f))
28	27	com12 14	. . 3 |- (h e. X_x e. (A \ B)C -> ((A.x e. A C =/= (/) /\ B C_ A /\ G e. X_x e. B C) -> E.f e. X_ x e. A CG C_ f))
29	28	19.23aiv 1674	. 2 |- (E.h h e. X_x e. (A \ B)C -> ((A.x e. A C =/= (/) /\ B C_ A /\ G e. X_x e. B C) -> E.f e. X_ x e. A CG C_ f))
30	16, 29	mpcom 60	1 |- ((A.x e. A C =/= (/) /\ B C_ A /\ G e. X_x e. B C) -> E.f e. X_ x e. A CG C_ f)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106   \ cdif 2590   u. cun 2591   C_ wss 2593  (/)c0 2875   |` cres 3988  X_cixp 5406

This theorem is referenced by:  prl1 14513

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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