Theorem prjnpl 14510

Description: Projection of nuple.

Assertion

Ref	Expression
prjnpl	|- ((B C_ A /\ F e. X_x e. A C) -> (F |` B) e. X_x e. B C)

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

Proof of Theorem prjnpl

Step	Hyp	Ref	Expression
1		elixp2 5408	. . 3 |- (F e. X_x e. A C <-> (F e. _V /\ F Fn A /\ A.x e. A (F` x) e. C))
2		fnssres 4526	. . . . . . . . 9 |- ((F Fn A /\ B C_ A) -> (F |` B) Fn B)
3		ssralv 2672	. . . . . . . . . . 11 |- (B C_ A -> (A.x e. A (F` x) e. C -> A.x e. B (F` x) e. C))
4		fvres 4691	. . . . . . . . . . . . . . . . . 18 |- (x e. B -> ((F |` B)` x) = (F` x))
5	4	eqcomd 1889	. . . . . . . . . . . . . . . . 17 |- (x e. B -> (F` x) = ((F |` B)` x))
6	5	eleq1d 1963	. . . . . . . . . . . . . . . 16 |- (x e. B -> ((F` x) e. C <-> ((F |` B)` x) e. C))
7	6	biimpd 170	. . . . . . . . . . . . . . 15 |- (x e. B -> ((F` x) e. C -> ((F |` B)` x) e. C))
8	7	ralimia 2166	. . . . . . . . . . . . . 14 |- (A.x e. B (F` x) e. C -> A.x e. B ((F |` B)` x) e. C)
9		elixp2 5408	. . . . . . . . . . . . . . . . 17 |- ((F |` B) e. X_x e. B C <-> ((F |` B) e. _V /\ (F |` B) Fn B /\ A.x e. B ((F |` B)` x) e. C))
10	9	biimpri 169	. . . . . . . . . . . . . . . 16 |- (((F |` B) e. _V /\ (F |` B) Fn B /\ A.x e. B ((F |` B)` x) e. C) -> (F |` B) e. X_x e. B C)
11	10	3exp 1066	. . . . . . . . . . . . . . 15 |- ((F |` B) e. _V -> ((F |` B) Fn B -> (A.x e. B ((F |` B)` x) e. C -> (F |` B) e. X_x e. B C)))
12	11	com3r 39	. . . . . . . . . . . . . 14 |- (A.x e. B ((F |` B)` x) e. C -> ((F |` B) e. _V -> ((F |` B) Fn B -> (F |` B) e. X_x e. B C)))
13	8, 12	syl 12	. . . . . . . . . . . . 13 |- (A.x e. B (F` x) e. C -> ((F |` B) e. _V -> ((F |` B) Fn B -> (F |` B) e. X_x e. B C)))
14		resexg 4250	. . . . . . . . . . . . 13 |- (F e. _V -> (F |` B) e. _V)
15	13, 14	syl5com 63	. . . . . . . . . . . 12 |- (F e. _V -> (A.x e. B (F` x) e. C -> ((F |` B) Fn B -> (F |` B) e. X_x e. B C)))
16	15	com3l 38	. . . . . . . . . . 11 |- (A.x e. B (F` x) e. C -> ((F |` B) Fn B -> (F e. _V -> (F |` B) e. X_x e. B C)))
17	3, 16	syl6 25	. . . . . . . . . 10 |- (B C_ A -> (A.x e. A (F` x) e. C -> ((F |` B) Fn B -> (F e. _V -> (F |` B) e. X_x e. B C))))
18	17	com3r 39	. . . . . . . . 9 |- ((F |` B) Fn B -> (B C_ A -> (A.x e. A (F` x) e. C -> (F e. _V -> (F |` B) e. X_x e. B C))))
19	2, 18	syl 12	. . . . . . . 8 |- ((F Fn A /\ B C_ A) -> (B C_ A -> (A.x e. A (F` x) e. C -> (F e. _V -> (F |` B) e. X_x e. B C))))
20	19	ex 402	. . . . . . 7 |- (F Fn A -> (B C_ A -> (B C_ A -> (A.x e. A (F` x) e. C -> (F e. _V -> (F |` B) e. X_x e. B C)))))
21	20	com13 37	. . . . . 6 |- (B C_ A -> (B C_ A -> (F Fn A -> (A.x e. A (F` x) e. C -> (F e. _V -> (F |` B) e. X_x e. B C)))))
22	21	pm2.43i 78	. . . . 5 |- (B C_ A -> (F Fn A -> (A.x e. A (F` x) e. C -> (F e. _V -> (F |` B) e. X_x e. B C))))
23	22	com14 42	. . . 4 |- (F e. _V -> (F Fn A -> (A.x e. A (F` x) e. C -> (B C_ A -> (F |` B) e. X_x e. B C))))
24	23	3imp 1061	. . 3 |- ((F e. _V /\ F Fn A /\ A.x e. A (F` x) e. C) -> (B C_ A -> (F |` B) e. X_x e. B C))
25	1, 24	sylbi 216	. 2 |- (F e. X_x e. A C -> (B C_ A -> (F |` B) e. X_x e. B C))
26	25	impcom 378	1 |- ((B C_ A /\ F e. X_x e. A C) -> (F |` B) e. X_x e. B C)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593   |` cres 3988   Fn wfn 3993  ` cfv 3998  X_cixp 5406

This theorem is referenced by:  prl 14512

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

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-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-fv 4014  df-ixp 5407

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