Theorem oprabvalig 4015

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 1531	. . . . . . . . . . 11 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 616	. . . . . . . . . 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 627	. . . . . . . . 9 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
5		eleq1 1531	. . . . . . . . . . 11 |- (y = B -> (y e. S <-> B e. S))
6	5	anbi2d 615	. . . . . . . . . 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 627	. . . . . . . . 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 615	. . . . . . . . 9 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
11	4, 8, 10	eloprabg 3998	. . . . . . . 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 417	. . . . . . 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 377	. . . . . 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 798	. . . 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 1427	. . . . . . 7 |- (E*z((x e. R /\ y e. S) /\ ph) <-> ((x e. R /\ y e. S) -> E*zph))
19	17, 18	mpbir 190	. . . . . 6 |- E*z((x e. R /\ y e. S) /\ ph)
20	19	funoprab 4002	. . . . 5 |- Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
21		funopfvg 3743	. . . . 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 695	. . . 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 801	. . 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 3956	. . . 4 |- (AFB) = (F` <.A, B>.)
26		oprabvalig.5	. . . . 5 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
27	26	fveq1i 3716	. . . 4 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
28	25, 27	eqtr 1492	. . 3 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
29	28	eqeq1i 1479	. 2 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
30	24, 29	syl6ibr 213	1 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  E*wmo 1379  <.cop 2407  Fun wfun 3171  ` cfv 3177  (class class class)co 3954  {copab2 3955

This theorem is referenced by:  oprabvali 4016  oprabval2gf 4017  spwval2 8595

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-opr 3956  df-oprab 3957

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