Theorem oprabval2gf 4026

Description: The value of an operation class abstraction. A version of oprabval2g 4027 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   x,B,y,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 1475	. . 3 |- S = S
2		visset 1813	. . . . . 6 |- v e. V
3		ax-17 971	. . . . . . 7 |- (u = A -> A.y u = A)
4		visset 1813	. . . . . . . . 9 |- u e. V
5	4	eqvinc 1883	. . . . . . . 8 |- (u = A <-> E.x(x = u /\ x = A))
6		ax-17 971	. . . . . . . . . . 11 |- (w e. u -> A.x w e. u)
7	4, 6	hbcsb1 2025	. . . . . . . . . 10 |- (w e. [_u / x]_R -> A.x w e. [_u / x]_R)
8		oprabval2gf.1	. . . . . . . . . 10 |- (w e. G -> A.x w e. G)
9	7, 8	hbeq 1565	. . . . . . . . 9 |- ([_u / x]_R = G -> A.x[_u / x]_R = G)
10		csbeq1a 2006	. . . . . . . . . 10 |- (x = u -> R = [_u / x]_R)
11		oprabval2gf.3	. . . . . . . . . 10 |- (x = A -> R = G)
12	10, 11	sylan9req 1528	. . . . . . . . 9 |- ((x = u /\ x = A) -> [_u / x]_R = G)
13	9, 12	19.23ai 1064	. . . . . . . 8 |- (E.x(x = u /\ x = A) -> [_u / x]_R = G)
14	5, 13	sylbi 199	. . . . . . 7 |- (u = A -> [_u / x]_R = G)
15	3, 14	csbeq2d 2018	. . . . . 6 |- ((u = A /\ v e. V) -> [_v / y]_[_u / x]_R = [_v / y]_G)
16	2, 15	mpan2 696	. . . . 5 |- (u = A -> [_v / y]_[_u / x]_R = [_v / y]_G)
17	16	eqeq2d 1486	. . . 4 |- (u = A -> (z = [_v / y]_[_u / x]_R <-> z = [_v / y]_G))
18	2	eqvinc 1883	. . . . . 6 |- (v = B <-> E.y(y = v /\ y = B))
19		ax-17 971	. . . . . . . . 9 |- (w e. v -> A.y w e. v)
20	2, 19	hbcsb1 2025	. . . . . . . 8 |- (w e. [_v / y]_G -> A.y w e. [_v / y]_G)
21		oprabval2gf.2	. . . . . . . 8 |- (w e. S -> A.y w e. S)
22	20, 21	hbeq 1565	. . . . . . 7 |- ([_v / y]_G = S -> A.y[_v / y]_G = S)
23		csbeq1a 2006	. . . . . . . 8 |- (y = v -> G = [_v / y]_G)
24		oprabval2gf.4	. . . . . . . 8 |- (y = B -> G = S)
25	23, 24	sylan9req 1528	. . . . . . 7 |- ((y = v /\ y = B) -> [_v / y]_G = S)
26	22, 25	19.23ai 1064	. . . . . 6 |- (E.y(y = v /\ y = B) -> [_v / y]_G = S)
27	18, 26	sylbi 199	. . . . 5 |- (v = B -> [_v / y]_G = S)
28	27	eqeq2d 1486	. . . 4 |- (v = B -> (z = [_v / y]_G <-> z = S))
29		eqeq1 1481	. . . 4 |- (z = S -> (z = S <-> S = S))
30		moeq 1920	. . . . 5 |- E*z z = [_v / y]_[_u / x]_R
31	30	a1i 8	. . . 4 |- ((u e. C /\ v e. D) -> E*z z = [_v / y]_[_u / x]_R)
32		eqid 1475	. . . 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)}
33	17, 28, 29, 31, 32	oprabvalig 4024	. . 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))
34	1, 33	mpi 44	. 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)
35		oprabval2gf.5	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}
36		ax-17 971	. . . . 5 |- (((x e. C /\ y e. D) /\ z = R) -> A.u((x e. C /\ y e. D) /\ z = R))
37		ax-17 971	. . . . 5 |- (((x e. C /\ y e. D) /\ z = R) -> A.v((x e. C /\ y e. D) /\ z = R))
38		ax-17 971	. . . . . 6 |- ((u e. C /\ v e. D) -> A.x(u e. C /\ v e. D))
39		ax-17 971	. . . . . . 7 |- (w e. z -> A.x w e. z)
40		ax-17 971	. . . . . . . . 9 |- (w e. v -> A.x w e. v)
41	40, 7	hbcsbg 2026	. . . . . . . 8 |- (v e. V -> (w e. [_v / y]_[_u / x]_R -> A.x w e. [_v / y]_[_u / x]_R))
42	2, 41	ax-mp 7	. . . . . . 7 |- (w e. [_v / y]_[_u / x]_R -> A.x w e. [_v / y]_[_u / x]_R)
43	39, 42	hbeq 1565	. . . . . 6 |- (z = [_v / y]_[_u / x]_R -> A.x z = [_v / y]_[_u / x]_R)
44	38, 43	hban 1009	. . . . 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))
45		ax-17 971	. . . . . 6 |- ((u e. C /\ v e. D) -> A.y(u e. C /\ v e. D))
46		ax-17 971	. . . . . . 7 |- (w e. z -> A.y w e. z)
47	2, 19	hbcsb1 2025	. . . . . . 7 |- (w e. [_v / y]_[_u / x]_R -> A.y w e. [_v / y]_[_u / x]_R)
48	46, 47	hbeq 1565	. . . . . 6 |- (z = [_v / y]_[_u / x]_R -> A.y z = [_v / y]_[_u / x]_R)
49	45, 48	hban 1009	. . . . 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))
50		eleq1 1534	. . . . . . . 8 |- (x = u -> (x e. C <-> u e. C))
51	50	anbi1d 617	. . . . . . 7 |- (x = u -> ((x e. C /\ y e. D) <-> (u e. C /\ y e. D)))
52	10	eqeq2d 1486	. . . . . . 7 |- (x = u -> (z = R <-> z = [_u / x]_R))
53	51, 52	anbi12d 628	. . . . . 6 |- (x = u -> (((x e. C /\ y e. D) /\ z = R) <-> ((u e. C /\ y e. D) /\ z = [_u / x]_R)))
54		eleq1 1534	. . . . . . . 8 |- (y = v -> (y e. D <-> v e. D))
55	54	anbi2d 616	. . . . . . 7 |- (y = v -> ((u e. C /\ y e. D) <-> (u e. C /\ v e. D)))
56		csbeq1a 2006	. . . . . . . 8 |- (y = v -> [_u / x]_R = [_v / y]_[_u / x]_R)
57	56	eqeq2d 1486	. . . . . . 7 |- (y = v -> (z = [_u / x]_R <-> z = [_v / y]_[_u / x]_R))
58	55, 57	anbi12d 628	. . . . . 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)))
59	53, 58	sylan9bb 540	. . . . 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)))
60	36, 37, 44, 49, 59	cbvoprab12 3998	. . . 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)}
61	35, 60	eqtr 1495	. . 3 |- F = {<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}
62	61	opreqi 3974	. 2 |- (AFB) = (A{<.<.u, v>., z>. | ((u e. C /\ v e. D) /\ z = [_v / y]_[_u / x]_R)}B)
63	34, 62	syl5eq 1519	1 |- ((A e. C /\ B e. D /\ S e. H) -> (AFB) = S)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  E*wmo 1381  Vcvv 1811  [_csb 2001  (class class class)co 3963  {copab2 3964

This theorem is referenced by:  oprabval2g 4027

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-sbc 1942  df-csb 2002  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-fv 3198  df-opr 3965