Theorem isprj1 14505

Description: Definition of a projection. I is a set of indices. P is a cartesian product.

Assertion

Ref	Expression
isprj1	|- ((P e. Q /\ I e. J) -> (P prj I) = {<.f, g>. | (f e. P /\ g = (f |` I))})

Distinct variable groups:   f,I,g   P,f,g

Proof of Theorem isprj1

Step	Hyp	Ref	Expression
1		elisset 2299	. . 3 |- (P e. Q -> P e. _V)
2		elisset 2299	. . 3 |- (I e. J -> I e. _V)
3	1, 2	anim12i 360	. 2 |- ((P e. Q /\ I e. J) -> (P e. _V /\ I e. _V))
4		opabex2g 4540	. . 3 |- (P e. Q -> {<.f, g>. | (f e. P /\ g = (f |` I))} e. _V)
5	4	adantr 425	. 2 |- ((P e. Q /\ I e. J) -> {<.f, g>. | (f e. P /\ g = (f |` I))} e. _V)
6		eleq2 1958	. . . . . 6 |- (x = P -> (f e. x <-> f e. P))
7	6	anbi1d 679	. . . . 5 |- (x = P -> ((f e. x /\ g = (f |` y)) <-> (f e. P /\ g = (f |` y))))
8	7	opabbidv 3401	. . . 4 |- (x = P -> {<.f, g>. | (f e. x /\ g = (f |` y))} = {<.f, g>. | (f e. P /\ g = (f |` y))})
9		reseq2 4219	. . . . . . 7 |- (y = I -> (f |` y) = (f |` I))
10	9	eqeq2d 1895	. . . . . 6 |- (y = I -> (g = (f |` y) <-> g = (f |` I)))
11	10	anbi2d 678	. . . . 5 |- (y = I -> ((f e. P /\ g = (f |` y)) <-> (f e. P /\ g = (f |` I))))
12	11	opabbidv 3401	. . . 4 |- (y = I -> {<.f, g>. | (f e. P /\ g = (f |` y))} = {<.f, g>. | (f e. P /\ g = (f |` I))})
13		df-prj 14492	. . . . 5 |- prj = {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}}
14		reloprab 4918	. . . . . 6 |- Rel {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}}
15		reldmoprab 4934	. . . . . 6 |- Rel dom {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}}
16		resid2 14425	. . . . . 6 |- ((Rel {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}} /\ Rel dom {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}}) -> ({<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}} |` (_V X. _V)) = {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}})
17	14, 15, 16	mp2an 761	. . . . 5 |- ({<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}} |` (_V X. _V)) = {<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}}
18		resoprab 4938	. . . . 5 |- ({<.<.x, y>., z>. | z = {<.f, g>. | (f e. x /\ g = (f |` y))}} |` (_V X. _V)) = {<.<.x, y>., z>. | ((x e. _V /\ y e. _V) /\ z = {<.f, g>. | (f e. x /\ g = (f |` y))})}
19	13, 17, 18	3eqtr2i 1915	. . . 4 |- prj = {<.<.x, y>., z>. | ((x e. _V /\ y e. _V) /\ z = {<.f, g>. | (f e. x /\ g = (f |` y))})}
20	8, 12, 19	oprabval2g 4956	. . 3 |- ((P e. _V /\ I e. _V /\ {<.f, g>. | (f e. P /\ g = (f |` I))} e. _V) -> (P prj I) = {<.f, g>. | (f e. P /\ g = (f |` I))})
21	20	3expia 1069	. 2 |- ((P e. _V /\ I e. _V) -> ({<.f, g>. | (f e. P /\ g = (f |` I))} e. _V -> (P prj I) = {<.f, g>. | (f e. P /\ g = (f |` I))}))
22	3, 5, 21	sylc 83	1 |- ((P e. Q /\ I e. J) -> (P prj I) = {<.f, g>. | (f e. P /\ g = (f |` I))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {copab 3395   X. cxp 3984  dom cdm 3986   |` cres 3988  Rel wrel 3991  (class class class)co 4884  {copab2 4885   prj cproj 14490

This theorem is referenced by:  isprj2 14506  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-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-prj 14492

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