Theorem prmapcp3 14498

Description: A projection is a mapping whose domain is a cartesian product.

Assertion

Ref	Expression
prmapcp3	|- ((P e. Q /\ I e. A) -> (P pr I) Fn P)

Proof of Theorem prmapcp3

Step	Hyp	Ref	Expression
1		ispr1 14496	. 2 |- ((P e. Q /\ I e. A) -> (P pr I) = {<.f, g>. | (f e. P /\ g = (f` I))})
2		fvex 4689	. . . 4 |- (f` I) e. _V
3		eqid 1884	. . . 4 |- {<.f, g>. | (f e. P /\ g = (f` I))} = {<.f, g>. | (f e. P /\ g = (f` I))}
4	2, 3	fnopab2 4549	. . 3 |- {<.f, g>. | (f e. P /\ g = (f` I))} Fn P
5		fneq1 4503	. . 3 |- ((P pr I) = {<.f, g>. | (f e. P /\ g = (f` I))} -> ((P pr I) Fn P <-> {<.f, g>. | (f e. P /\ g = (f` I))} Fn P))
6	4, 5	mpbiri 211	. 2 |- ((P pr I) = {<.f, g>. | (f e. P /\ g = (f` I))} -> (P pr I) Fn P)
7	1, 6	syl 12	1 |- ((P e. Q /\ I e. A) -> (P pr I) Fn P)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {copab 3395   Fn wfn 3993  ` cfv 3998  (class class class)co 4884   pr cpro 14489

This theorem is referenced by:  npincppr 14501

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-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-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-opr 4886  df-oprab 4887  df-pro 14491

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