Theorem prjmapcp 14507

Description: A projection is a mapping from a cartesian product to one of its restriction.

Assertion

Ref	Expression
prjmapcp	|- ((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)

Distinct variable groups:   x,A   x,I

Proof of Theorem prjmapcp

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . 6 |- f e. _V
2		resexg 4250	. . . . . 6 |- (f e. _V -> (f |` I) e. _V)
3	1, 2	ax-mp 7	. . . . 5 |- (f |` I) e. _V
4	3	elixp 5409	. . . 4 |- ((f |` I) e. X_x e. I B <-> ((f |` I) Fn I /\ A.x e. I ((f |` I)` x) e. B))
5		elixp2 5408	. . . . . . . 8 |- (f e. X_x e. A B <-> (f e. _V /\ f Fn A /\ A.x e. A (f` x) e. B))
6		fnssres 4526	. . . . . . . . . 10 |- ((f Fn A /\ I C_ A) -> (f |` I) Fn I)
7	6	ex 402	. . . . . . . . 9 |- (f Fn A -> (I C_ A -> (f |` I) Fn I))
8	7	3ad2ant2 898	. . . . . . . 8 |- ((f e. _V /\ f Fn A /\ A.x e. A (f` x) e. B) -> (I C_ A -> (f |` I) Fn I))
9	5, 8	sylbi 216	. . . . . . 7 |- (f e. X_x e. A B -> (I C_ A -> (f |` I) Fn I))
10	9	com12 14	. . . . . 6 |- (I C_ A -> (f e. X_x e. A B -> (f |` I) Fn I))
11	10	3ad2ant1 897	. . . . 5 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> (f e. X_x e. A B -> (f |` I) Fn I))
12	11	imp 377	. . . 4 |- (((I C_ A /\ A e. C /\ A.x e. A B e. D) /\ f e. X_x e. A B) -> (f |` I) Fn I)
13		pm2.27 76	. . . . . . . . . 10 |- (A.x e. A (f` x) e. B -> ((A.x e. A (f` x) e. B -> A.x e. I (f` x) e. B) -> A.x e. I (f` x) e. B))
14		fvres 4691	. . . . . . . . . . . . . 14 |- (x e. I -> ((f |` I)` x) = (f` x))
15	14	eqcomd 1889	. . . . . . . . . . . . 13 |- (x e. I -> (f` x) = ((f |` I)` x))
16	15	eleq1d 1963	. . . . . . . . . . . 12 |- (x e. I -> ((f` x) e. B <-> ((f |` I)` x) e. B))
17	16	biimpd 170	. . . . . . . . . . 11 |- (x e. I -> ((f` x) e. B -> ((f |` I)` x) e. B))
18	17	ralimia 2166	. . . . . . . . . 10 |- (A.x e. I (f` x) e. B -> A.x e. I ((f |` I)` x) e. B)
19	13, 18	syl6 25	. . . . . . . . 9 |- (A.x e. A (f` x) e. B -> ((A.x e. A (f` x) e. B -> A.x e. I (f` x) e. B) -> A.x e. I ((f |` I)` x) e. B))
20	19	adantl 424	. . . . . . . 8 |- ((f Fn A /\ A.x e. A (f` x) e. B) -> ((A.x e. A (f` x) e. B -> A.x e. I (f` x) e. B) -> A.x e. I ((f |` I)` x) e. B))
21		ssralv 2672	. . . . . . . 8 |- (I C_ A -> (A.x e. A (f` x) e. B -> A.x e. I (f` x) e. B))
22	20, 21	syl5com 63	. . . . . . 7 |- (I C_ A -> ((f Fn A /\ A.x e. A (f` x) e. B) -> A.x e. I ((f |` I)` x) e. B))
23	22	3ad2ant1 897	. . . . . 6 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> ((f Fn A /\ A.x e. A (f` x) e. B) -> A.x e. I ((f |` I)` x) e. B))
24	1	elixp 5409	. . . . . 6 |- (f e. X_x e. A B <-> (f Fn A /\ A.x e. A (f` x) e. B))
25	23, 24	syl5ib 223	. . . . 5 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> (f e. X_x e. A B -> A.x e. I ((f |` I)` x) e. B))
26	25	imp 377	. . . 4 |- (((I C_ A /\ A e. C /\ A.x e. A B e. D) /\ f e. X_x e. A B) -> A.x e. I ((f |` I)` x) e. B)
27	4, 12, 26	sylanbrc 527	. . 3 |- (((I C_ A /\ A e. C /\ A.x e. A B e. D) /\ f e. X_x e. A B) -> (f |` I) e. X_x e. I B)
28	27	r19.21aiva 2176	. 2 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> A.f e. X_ x e. A B(f |` I) e. X_x e. I B)
29		eqid 1884	. . . . 5 |- X_x e. A B = X_x e. A B
30	29	isprj2 14506	. . . 4 |- ((I e. _V /\ A e. C /\ A.x e. A B e. D) -> (X_x e. A B prj I) = {<.f, y>. | (f e. X_x e. A B /\ y = (f |` I))})
31		ssexg 3457	. . . . 5 |- ((I C_ A /\ A e. C) -> I e. _V)
32	31	3adant3 896	. . . 4 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> I e. _V)
33	30, 32	syld3an1 1143	. . 3 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> (X_x e. A B prj I) = {<.f, y>. | (f e. X_x e. A B /\ y = (f |` I))})
34		fopab2g 14485	. . 3 |- ((X_x e. A B prj I) = {<.f, y>. | (f e. X_x e. A B /\ y = (f |` I))} -> (A.f e. X_ x e. A B(f |` I) e. X_x e. I B <-> (X_x e. A B prj I):X_x e. A B-->X_x e. I B))
35	33, 34	syl 12	. 2 |- ((I C_ A /\ A e. C /\ A.x e. A B e. D) -> (A.f e. X_ x e. A B(f |` I) e. X_x e. I B <-> (X_x e. A B prj I):X_x e. A B-->X_x e. I B))
36	28, 35	mpbid 212	1 |- ((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)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  {copab 3395   |` cres 3988   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  X_cixp 5406   prj cproj 14490

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

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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-iun 3257  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-fv 4014  df-opr 4886  df-oprab 4887  df-ixp 5407  df-prj 14492

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