Theorem oprabvalig 4082

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 1581	. . . . . . . . . . 11 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 628	. . . . . . . . . 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 639	. . . . . . . . 9 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
5		eleq1 1581	. . . . . . . . . . 11 |- (y = B -> (y e. S <-> B e. S))
6	5	anbi2d 627	. . . . . . . . . 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 639	. . . . . . . . 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 627	. . . . . . . . 9 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
11	4, 8, 10	eloprabg 4065	. . . . . . . 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 426	. . . . . . 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 386	. . . . . 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 11	. . . . 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 811	. . . 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 64	. . 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 1471	. . . . . . 7 |- (E*z((x e. R /\ y e. S) /\ ph) <-> ((x e. R /\ y e. S) -> E*zph))
19	17, 18	mpbir 197	. . . . . 6 |- E*z((x e. R /\ y e. S) /\ ph)
20	19	funoprab 4069	. . . . 5 |- Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
21		funopfvg 3809	. . . . 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 708	. . . 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 814	. . 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 27	. 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 4023	. . . 4 |- (AFB) = (F` <.A, B>.)
26		oprabvalig.5	. . . . 5 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
27	26	fveq1i 3782	. . . 4 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
28	25, 27	eqtri 1542	. . 3 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
29	28	eqeq1i 1529	. 2 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
30	24, 29	syl6ibr 220	1 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   /\ w3a 787   = wceq 997   e. wcel 999  E*wmo 1423  <.cop 2463  Fun wfun 3233  ` cfv 3239  (class class class)co 4021  {copab2 4022

This theorem is referenced by:  oprabvali 4083  oprabval2gf 4084  spwval2 8737

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fv 3255  df-opr 4023  df-oprab 4024

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