Theorem oprabval 4023

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 1534	. . . . . . . 8 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 617	. . . . . . 7 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
3		eleq1 1534	. . . . . . . 8 |- (y = B -> (y e. S <-> B e. S))
4	3	anbi2d 616	. . . . . . 7 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
5	2, 4	opelopabg 2817	. . . . . 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 593	. . . . 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 4013	. . . . . 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 3754	. . . . . 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 695	. . . . 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 628	. . . . . 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 628	. . . . . 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 616	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
19	14, 16, 18	eloprabg 4007	. . . . 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 905	. . . 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 528	. . 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 3965	. . . . 5 |- (AFB) = (F` <.A, B>.)
23		oprabval.6	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
24	23	fveq1i 3725	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
25	22, 24	eqtr 1495	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
26	25	eqeq1i 1482	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
27	21, 26	syl5bb 532	. 2 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
28	27	bianabs 653	1 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E!weu 1380  Vcvv 1811  <.cop 2411  {copab 2666   Fn wfn 3177  ` cfv 3182  (class class class)co 3963  {copab2 3964

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-opr 3965  df-oprab 3966

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