Theorem oprabval 3047

Description: The value of an operation 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 group(s):   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 1149	. . . . . . . . 9 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 469	. . . . . . . 8 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
3		eleq1 1149	. . . . . . . . 9 |- (y = B -> (y e. S <-> B e. S))
4	3	anbi2d 468	. . . . . . . 8 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
5	2, 4	opelopabg 2115	. . . . . . 7 |- ((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	biimprd 136	. . . . . 6 |- ((A e. R /\ B e. S) -> ((A e. R /\ B e. S) -> <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)}))
7	6	pm2.43i 58	. . . . 5 |- ((A e. R /\ B e. S) -> <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)})
8		oprabval.5	. . . . . . 7 |- ((x e. R /\ y e. S) -> E!zph)
9	8	fnoprab 3038	. . . . . 6 |- {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)}
10		oprabval.1	. . . . . . 7 |- C e. V
11	10	fnfvop 2856	. . . . . 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)}))
12	9, 11	mpan 518	. . . . 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)}))
13	7, 12	syl 12	. . . 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)}))
14		oprabval.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
15	2, 14	anbi12d 476	. . . . . 6 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
16		oprabval.3	. . . . . . 7 |- (y = B -> (ps <-> ch))
17	4, 16	anbi12d 476	. . . . . 6 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
18		oprabval.4	. . . . . . 7 |- (z = C -> (ch <-> th))
19	18	anbi2d 468	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
20	15, 17, 19	eloprabg 3035	. . . . 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)))
21	10, 20	mp3an3 641	. . . 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)))
22	13, 21	bitrd 406	. . 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)))
23		df-opr 3003	. . . . 5 |- (AFB) = (F` <.A, B>.)
24		oprabval.6	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
25	24	fveq1i 2833	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
26	23, 25	eqtr 1119	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
27	26	cleq1i 1108	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
28	22, 27	syl5bb 410	. 2 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
29		ibar 487	. 2 |- ((A e. R /\ B e. S) -> (th <-> ((A e. R /\ B e. S) /\ th)))
30	28, 29	bitr4d 409	1 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810  {copab 2055   Fn wfn 2417  ` cfv 2422  (class class class)co 3001  {copab2 3002

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-opr 3003  df-oprab 3004

metamath.org