Theorem oprabval 4952

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		oprabval.1	. . . . . 6 |- C e. _V
2	1	fnopfvb 4713	. . . . 5 |- (({<.<.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)}))
3		oprabval.5	. . . . . 6 |- ((x e. R /\ y e. S) -> E!zph)
4	3	fnoprab 4942	. . . . 5 |- {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)}
5		eleq1 1957	. . . . . . . 8 |- (x = A -> (x e. R <-> A e. R))
6	5	anbi1d 679	. . . . . . 7 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
7		eleq1 1957	. . . . . . . 8 |- (y = B -> (y e. S <-> B e. S))
8	7	anbi2d 678	. . . . . . 7 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
9	6, 8	opelopabg 3567	. . . . . 6 |- ((A e. R /\ B e. S) -> (<.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)} <-> (A e. R /\ B e. S)))
10	9	ibir 653	. . . . 5 |- ((A e. R /\ B e. S) -> <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)})
11	2, 4, 10	sylancr 526	. . . 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)}))
12		oprabval.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
13	6, 12	anbi12d 690	. . . . . 6 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
14		oprabval.3	. . . . . . 7 |- (y = B -> (ps <-> ch))
15	8, 14	anbi12d 690	. . . . . 6 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
16		oprabval.4	. . . . . . 7 |- (z = C -> (ch <-> th))
17	16	anbi2d 678	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
18	13, 15, 17	eloprabg 4936	. . . . 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)))
19	1, 18	mp3an3 1180	. . . 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)))
20	11, 19	bitrd 587	. . 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)))
21		df-opr 4886	. . . . 5 |- (AFB) = (F` <.A, B>.)
22		oprabval.6	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
23	22	fveq1i 4682	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
24	21, 23	eqtri 1908	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
25	24	eqeq1i 1891	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
26	20, 25	syl5bb 591	. 2 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
27	26	bianabs 715	1 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  _Vcvv 2292  <.cop 3046  {copab 3395   Fn wfn 3993  ` cfv 3998  (class class class)co 4884  {copab2 4885

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887

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