Theorem oprabval 4029

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 1537	. . . . . . . 8 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 619	. . . . . . 7 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
3		eleq1 1537	. . . . . . . 8 |- (y = B -> (y e. S <-> B e. S))
4	3	anbi2d 618	. . . . . . 7 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
5	2, 4	opelopabg 2823	. . . . . 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 595	. . . . 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 4019	. . . . . 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 3760	. . . . . 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 697	. . . . 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 630	. . . . . 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 630	. . . . . 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 618	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
19	14, 16, 18	eloprabg 4013	. . . . 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 907	. . . 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 530	. . 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 3971	. . . . 5 |- (AFB) = (F` <.A, B>.)
23		oprabval.6	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
24	23	fveq1i 3731	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
25	22, 24	eqtr 1498	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
26	25	eqeq1i 1485	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
27	21, 26	syl5bb 534	. 2 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
28	27	bianabs 655	1 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E!weu 1382  Vcvv 1814  <.cop 2415  {copab 2671   Fn wfn 3183  ` cfv 3188  (class class class)co 3969  {copab2 3970

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971  df-oprab 3972

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