Theorem oprabval 4008

Description: The value of an operation class abstraction.

Hypotheses

Ref	Expression
oprabval.1	|- C e. V
oprabval.2	|- (x = A -> (ph <-> ps))
oprabval.3	|- (y = B -> (ps <-> ch))
oprabval.4	|- (z = C -> (ch <-> th))
oprabval.5	|- ((x e. R /\ y e. S) -> E!zph)
oprabval.6	|- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}

Assertion

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

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 oprabval

Step	Hyp	Ref	Expression
1		eleq1 1526	. . . . . . . 8 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 615	. . . . . . 7 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
3		eleq1 1526	. . . . . . . 8 |- (y = B -> (y e. S <-> B e. S))
4	3	anbi2d 614	. . . . . . 7 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
5	2, 4	opelopabg 2806	. . . . . 6 |- ((A e. R /\ B e. S) -> (<.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)} <-> (A e. R /\ B e. S)))
6	5	ibir 591	. . . . 5 |- ((A e. R /\ B e. S) -> <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)})
7		oprabval.5	. . . . . . 7 |- ((x e. R /\ y e. S) -> E!zph)
8	7	fnoprab 3998	. . . . . 6 |- {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)}
9		oprabval.1	. . . . . . 7 |- C e. V
10	9	fnopfvb 3739	. . . . . 6 |- (({<.<.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)}))
11	8, 10	mpan 693	. . . . 5 |- (<.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)}))
12	6, 11	syl 10	. . . 4 |- ((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)}))
13		oprabval.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
14	2, 13	anbi12d 626	. . . . . 6 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
15		oprabval.3	. . . . . . 7 |- (y = B -> (ps <-> ch))
16	4, 15	anbi12d 626	. . . . . 6 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
17		oprabval.4	. . . . . . 7 |- (z = C -> (ch <-> th))
18	17	anbi2d 614	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
19	14, 16, 18	eloprabg 3992	. . . . 5 |- ((A e. R /\ B e. S /\ C e. V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
20	9, 19	mp3an3 902	. . . 4 |- ((A e. R /\ B e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
21	12, 20	bitrd 526	. . 3 |- ((A e. R /\ B e. S) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> ((A e. R /\ B e. S) /\ th)))
22		df-opr 3950	. . . . 5 |- (AFB) = (F` <.A, B>.)
23		oprabval.6	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
24	23	fveq1i 3710	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
25	22, 24	eqtr 1487	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
26	25	eqeq1i 1474	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
27	21, 26	syl5bb 530	. 2 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
28	27	bianabs 651	1 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E!weu 1373  Vcvv 1802  <.cop 2401  {copab 2656   Fn wfn 3167  ` cfv 3172  (class class class)co 3948  {copab2 3949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-opr 3950  df-oprab 3951

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