Theorem oprabval2gf 4955

Description: The value of an operation class abstraction. A version of oprabval2g 4956 using bound-variable hypotheses.

Hypotheses

Ref	Expression
oprabval2gf.1	|- (w e. G -> A.x w e. G)
oprabval2gf.2	|- (w e. S -> A.y w e. S)
oprabval2gf.3	|- (x = A -> R = G)
oprabval2gf.4	|- (y = B -> G = S)
oprabval2gf.5	|- F = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}

Assertion

Ref	Expression
oprabval2gf	|- ((A e. C /\ B e. D /\ S e. H) -> (AFB) = S)

Distinct variable groups:   x,y,z,A   y,B,z   x,C,y,z   w,G   x,D,y,z   z,w,R   w,S,z   x,w,y

Proof of Theorem oprabval2gf

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- S = S
2		visset 2295	. . . . . 6 |- v e. _V
3		ax-17 1317	. . . . . . 7 |- (u = A -> A.y u = A)
4		ax-17 1317	. . . . . . . 8 |- (v e. A -> A.x v e. A)
5		oprabval2gf.1	. . . . . . . . 9 |- (w e. G -> A.x w e. G)
6	5	hblem 1993	. . . . . . . 8 |- (v e. G -> A.x v e. G)
7		oprabval2gf.3	. . . . . . . 8 |- (x = A -> R = G)
8	4, 6, 7	csbhypf 2572	. . . . . . 7 |- (u = A -> [_u / x]_R = G)
9	3, 8	csbeq2d 2561	. . . . . 6 |- ((u = A /\ v e. _V) -> [_v / y]_[_u / x]_R = [_v / y]_G)
10	2, 9	mpan2 760	. . . . 5 |- (u = A -> [_v / y]_[_u / x]_R = [_v / y]_G)
11	10	eqeq2d 1895	. . . 4 |- (u = A -> (z = [_v / y]_[_u / x]_R <-> z = [_v / y]_G))
12		ax-17 1317	. . . . . 6 |- (u e. B -> A.y u e. B)
13		oprabval2gf.2	. . . . . . 7 |- (w e. S -> A.y w e. S)
14	13	hblem 1993	. . . . . 6 |- (u e. S -> A.y u e. S)
15		oprabval2gf.4	. . . . . 6 |- (y = B -> G = S)
16	12, 14, 15	csbhypf 2572	. . . . 5 |- (v = B -> [_v / y]_G = S)
17	16	eqeq2d 1895	. . . 4 |- (v = B -> (z = [_v / y]_G <-> z = S))
18		eqeq1 1890	. . . 4 |- (z = S -> (z = S <-> S = S))
19		moeq 2431	. . . . 5 |- E*z z = [_v / y]_[_u / x]_R
20	19	a1i 8	. . . 4 |- ((u e. C /\ v e. D) -> E*z z = [_v / y]_[_u / x]_R)
21		eqid 1884	. . . 4 |- {<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)} = {<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}
22	11, 17, 18, 20, 21	oprabvalig 4953	. . 3 |- ((A e. C /\ B e. D /\ S e. H) -> (S = S -> (A{<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}B) = S))
23	1, 22	mpi 55	. 2 |- ((A e. C /\ B e. D /\ S e. H) -> (A{<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}B) = S)
24		oprabval2gf.5	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}
25		ax-17 1317	. . . . 5 |- (((x e. C /\ y e. D) /\ z = R) -> A.u((x e. C /\ y e. D) /\ z = R))
26		ax-17 1317	. . . . 5 |- (((x e. C /\ y e. D) /\ z = R) -> A.v((x e. C /\ y e. D) /\ z = R))
27		ax-17 1317	. . . . . 6 |- ((u e. C /\ v e. D) -> A.x(u e. C /\ v e. D))
28		ax-17 1317	. . . . . . 7 |- (w e. z -> A.x w e. z)
29		ax-17 1317	. . . . . . . . 9 |- (w e. v -> A.x w e. v)
30		visset 2295	. . . . . . . . . 10 |- u e. _V
31		ax-17 1317	. . . . . . . . . 10 |- (w e. u -> A.x w e. u)
32	30, 31	hbcsb1 2568	. . . . . . . . 9 |- (w e. [_u / x]_R -> A.x w e. [_u / x]_R)
33	29, 32	hbcsbg 2569	. . . . . . . 8 |- (v e. _V -> (w e. [_v / y]_[_u / x]_R -> A.x w e. [_v / y]_[_u / x]_R))
34	2, 33	ax-mp 7	. . . . . . 7 |- (w e. [_v / y]_[_u / x]_R -> A.x w e. [_v / y]_[_u / x]_R)
35	28, 34	hbeq 1995	. . . . . 6 |- (z = [_v / y]_[_u / x]_R -> A.x z = [_v / y]_[_u / x]_R)
36	27, 35	hban 1356	. . . . 5 |- (((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R) -> A.x((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R))
37		ax-17 1317	. . . . . 6 |- ((u e. C /\ v e. D) -> A.y(u e. C /\ v e. D))
38		ax-17 1317	. . . . . . 7 |- (w e. z -> A.y w e. z)
39		ax-17 1317	. . . . . . . 8 |- (w e. v -> A.y w e. v)
40	2, 39	hbcsb1 2568	. . . . . . 7 |- (w e. [_v / y]_[_u / x]_R -> A.y w e. [_v / y]_[_u / x]_R)
41	38, 40	hbeq 1995	. . . . . 6 |- (z = [_v / y]_[_u / x]_R -> A.y z = [_v / y]_[_u / x]_R)
42	37, 41	hban 1356	. . . . 5 |- (((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R) -> A.y((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R))
43		eleq1 1957	. . . . . . . 8 |- (x = u -> (x e. C <-> u e. C))
44	43	anbi1d 679	. . . . . . 7 |- (x = u -> ((x e. C /\ y e. D) <-> (u e. C /\ y e. D)))
45		csbeq1a 2546	. . . . . . . 8 |- (x = u -> R = [_u / x]_R)
46	45	eqeq2d 1895	. . . . . . 7 |- (x = u -> (z = R <-> z = [_u / x]_R))
47	44, 46	anbi12d 690	. . . . . 6 |- (x = u -> (((x e. C /\ y e. D) /\ z = R) <-> ((u e. C /\ y e. D) /\ z = [_u / x]_R)))
48		eleq1 1957	. . . . . . . 8 |- (y = v -> (y e. D <-> v e. D))
49	48	anbi2d 678	. . . . . . 7 |- (y = v -> ((u e. C /\ y e. D) <-> (u e. C /\ v e. D)))
50		csbeq1a 2546	. . . . . . . 8 |- (y = v -> [_u / x]_R = [_v / y]_[_u / x]_R)
51	50	eqeq2d 1895	. . . . . . 7 |- (y = v -> (z = [_u / x]_R <-> z = [_v / y]_[_u / x]_R))
52	49, 51	anbi12d 690	. . . . . 6 |- (y = v -> (((u e. C /\ y e. D) /\ z = [_u / x]_R) <-> ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)))
53	47, 52	sylan9bb 599	. . . . 5 |- ((x = u /\ y = v) -> (((x e. C /\ y e. D) /\ z = R) <-> ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)))
54	25, 26, 36, 42, 53	cbvoprab12 4926	. . . 4 |- {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)} = {<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}
55	24, 54	eqtri 1908	. . 3 |- F = {<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}
56	55	opreqi 4896	. 2 |- (AFB) = (A{<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}B)
57	23, 56	syl5eq 1940	1 |- ((A e. C /\ B e. D /\ S e. H) -> (AFB) = S)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E*wmo 1772  _Vcvv 2292  [_csb 2540  (class class class)co 4884  {copab2 4885

This theorem is referenced by:  oprabval2g 4956  oprpiece1res1 15880  oprpiece1res2 15881  cnresoprab 15915

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-sbc 2454  df-csb 2541  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-fv 4014  df-opr 4886  df-oprab 4887

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