Theorem oprabvalig 4953

Description: The value of an operation class abstraction (weak version).

Hypotheses

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

Assertion

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

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

Proof of Theorem oprabvalig

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . . . . . . 11 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 679	. . . . . . . . . 10 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
3		oprabvalig.1	. . . . . . . . . 10 |- (x = A -> (ph <-> ps))
4	2, 3	anbi12d 690	. . . . . . . . 9 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
5		eleq1 1957	. . . . . . . . . . 11 |- (y = B -> (y e. S <-> B e. S))
6	5	anbi2d 678	. . . . . . . . . 10 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
7		oprabvalig.2	. . . . . . . . . 10 |- (y = B -> (ps <-> ch))
8	6, 7	anbi12d 690	. . . . . . . . 9 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
9		oprabvalig.3	. . . . . . . . . 10 |- (z = C -> (ch <-> th))
10	9	anbi2d 678	. . . . . . . . 9 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
11	4, 8, 10	eloprabg 4936	. . . . . . . 8 |- ((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)))
12	11	biimpar 461	. . . . . . 7 |- (((A e. R /\ B e. S /\ C e. D) /\ ((A e. R /\ B e. S) /\ th)) -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})
13	12	exp32 408	. . . . . 6 |- ((A e. R /\ B e. S /\ C e. D) -> ((A e. R /\ B e. S) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
14	13	com12 14	. . . . 5 |- ((A e. R /\ B e. S) -> ((A e. R /\ B e. S /\ C e. D) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
15	14	3adant3 896	. . . 4 |- ((A e. R /\ B e. S /\ C e. D) -> ((A e. R /\ B e. S /\ C e. D) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
16	15	pm2.43i 78	. . 3 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
17		oprabvalig.4	. . . . . . 7 |- ((x e. R /\ y e. S) -> E*zph)
18		moanimv 1829	. . . . . . 7 |- (E*z((x e. R /\ y e. S) /\ ph) <-> ((x e. R /\ y e. S) -> E*zph))
19	17, 18	mpbir 207	. . . . . 6 |- E*z((x e. R /\ y e. S) /\ ph)
20	19	funoprab 4940	. . . . 5 |- Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
21		funopfvg 4711	. . . . 5 |- ((C e. D /\ Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}) -> (<.<.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))
22	20, 21	mpan2 760	. . . 4 |- (C e. D -> (<.<.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))
23	22	3ad2ant3 899	. . 3 |- ((A e. R /\ B e. S /\ C e. D) -> (<.<.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))
24	16, 23	syld 30	. 2 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
25		df-opr 4886	. . . 4 |- (AFB) = (F` <.A, B>.)
26		oprabvalig.5	. . . . 5 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
27	26	fveq1i 4682	. . . 4 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
28	25, 27	eqtri 1908	. . 3 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
29	28	eqeq1i 1891	. 2 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
30	24, 29	syl6ibr 230	1 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E*wmo 1772  <.cop 3046  Fun wfun 3992  ` cfv 3998  (class class class)co 4884  {copab2 4885

This theorem is referenced by:  oprabvali 4954  oprabval2gf 4955  spwval2 9996

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-fv 4014  df-opr 4886  df-oprab 4887

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