Theorem oprabvalg 15706

Description: The value of an operation class abstraction.

Hypotheses

Ref	Expression
oprabvalg.1	|- (x = A -> (ph <-> ps))
oprabvalg.2	|- (y = B -> (ps <-> ch))
oprabvalg.3	|- (z = C -> (ch <-> th))
oprabvalg.4	|- ((ta /\ (x e. R /\ y e. S)) -> E!zph)
oprabvalg.5	|- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}

Assertion

Ref	Expression
oprabvalg	|- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> ((AFB) = C <-> th))

Distinct variable groups:   ps,x   ch,x,y   th,x,y,z   ta,x,y   x,R,y,z   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z

Proof of Theorem oprabvalg

Step	Hyp	Ref	Expression
1		eqeq2 1893	. . . . . . . . . 10 |- (c = C -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = c <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
2		opeq2 3159	. . . . . . . . . . 11 |- (c = C -> <.<.A, B>., c>. = <.<.A, B>., C>.)
3	2	eleq1d 1963	. . . . . . . . . 10 |- (c = C -> (<.<.A, B>., c>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
4	1, 3	bibi12d 691	. . . . . . . . 9 |- (c = C -> ((({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = c <-> <.<.A, B>., c>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}) <-> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
5	4	imbi2d 674	. . . . . . . 8 |- (c = C -> (((ta /\ (A e. R /\ B e. S)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = c <-> <.<.A, B>., c>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})) <-> ((ta /\ (A e. R /\ B e. S)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))))
6		visset 2295	. . . . . . . . . 10 |- c e. _V
7	6	fnopfvb 4713	. . . . . . . . 9 |- (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)} /\ <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)}) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = c <-> <.<.A, B>., c>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
8		oprabvalg.4	. . . . . . . . . . . 12 |- ((ta /\ (x e. R /\ y e. S)) -> E!zph)
9	8	ex 402	. . . . . . . . . . 11 |- (ta -> ((x e. R /\ y e. S) -> E!zph))
10	9	19.21aivv 1665	. . . . . . . . . 10 |- (ta -> A.xA.y((x e. R /\ y e. S) -> E!zph))
11		fnoprabg 4941	. . . . . . . . . 10 |- (A.xA.y((x e. R /\ y e. S) -> E!zph) -> {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)})
12	10, 11	syl 12	. . . . . . . . 9 |- (ta -> {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)})
13		eleq1 1957	. . . . . . . . . . . 12 |- (x = A -> (x e. R <-> A e. R))
14	13	anbi1d 679	. . . . . . . . . . 11 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
15		eleq1 1957	. . . . . . . . . . . 12 |- (y = B -> (y e. S <-> B e. S))
16	15	anbi2d 678	. . . . . . . . . . 11 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
17	14, 16	opelopabg 3567	. . . . . . . . . 10 |- ((A e. R /\ B e. S) -> (<.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)} <-> (A e. R /\ B e. S)))
18	17	ibir 653	. . . . . . . . 9 |- ((A e. R /\ B e. S) -> <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)})
19	7, 12, 18	syl2an 503	. . . . . . . 8 |- ((ta /\ (A e. R /\ B e. S)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = c <-> <.<.A, B>., c>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
20	5, 19	vtoclg 2346	. . . . . . 7 |- (C e. D -> ((ta /\ (A e. R /\ B e. S)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
21	20	com12 14	. . . . . 6 |- ((ta /\ (A e. R /\ B e. S)) -> (C e. D -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
22	21	exp32 408	. . . . 5 |- (ta -> (A e. R -> (B e. S -> (C e. D -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))))
23	22	3imp2 1083	. . . 4 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
24		oprabvalg.1	. . . . . . 7 |- (x = A -> (ph <-> ps))
25	14, 24	anbi12d 690	. . . . . 6 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
26		oprabvalg.2	. . . . . . 7 |- (y = B -> (ps <-> ch))
27	16, 26	anbi12d 690	. . . . . 6 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
28		oprabvalg.3	. . . . . . 7 |- (z = C -> (ch <-> th))
29	28	anbi2d 678	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
30	25, 27, 29	eloprabg 4936	. . . . 5 |- ((A e. R /\ B e. S /\ C e. D) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
31	30	adantl 424	. . . 4 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
32	23, 31	bitrd 587	. . 3 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> ((A e. R /\ B e. S) /\ th)))
33		df-opr 4886	. . . . 5 |- (AFB) = (F` <.A, B>.)
34		oprabvalg.5	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
35	34	fveq1i 4682	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
36	33, 35	eqtri 1908	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
37	36	eqeq1i 1891	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
38	32, 37	syl5bb 591	. 2 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
39		biidd 188	. . . . 5 |- ((A e. R /\ B e. S) -> (((A e. R /\ B e. S) /\ th) <-> ((A e. R /\ B e. S) /\ th)))
40	39	bianabs 715	. . . 4 |- ((A e. R /\ B e. S) -> (((A e. R /\ B e. S) /\ th) <-> th))
41	40	3adant3 896	. . 3 |- ((A e. R /\ B e. S /\ C e. D) -> (((A e. R /\ B e. S) /\ th) <-> th))
42	41	adantl 424	. 2 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (((A e. R /\ B e. S) /\ th) <-> th))
43	38, 42	bitrd 587	1 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E!weu 1771  <.cop 3046  {copab 3395   Fn wfn 3993  ` cfv 3998  (class class class)co 4884  {copab2 4885

This theorem is referenced by:  eroprv 15734

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-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-opr 4886  df-oprab 4887

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