Theorem oprabval4g 3053

Description: Value of an operation given by an ordered pair abstraction. ( This is the operation analog of fvopab2 2878.)

Hypothesis

Ref	Expression
oprabval4g.1	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}

Assertion

Ref	Expression
oprabval4g	|- ((x e. A /\ y e. B /\ C e. D) -> (xFy) = C)

Distinct variable group(s):   x,y,z,A   x,B,y,z   z,C

Proof of Theorem oprabval4g

Step	Hyp	Ref	Expression
1		sbab 1188	. . . . 5 |- (x = w -> {u | [v / y]u e. C} = {f | [w / x]f e. {u | [v / y]u e. C}})
2	1	cleqcomd 1106	. . . 4 |- (x = w -> {f | [w / x]f e. {u | [v / y]u e. C}} = {u | [v / y]u e. C})
3	2	cleqcoms 1104	. . 3 |- (w = x -> {f | [w / x]f e. {u | [v / y]u e. C}} = {u | [v / y]u e. C})
4		sbab 1188	. . . . 5 |- (y = v -> C = {u | [v / y]u e. C})
5	4	cleqcomd 1106	. . . 4 |- (y = v -> {u | [v / y]u e. C} = C)
6	5	cleqcoms 1104	. . 3 |- (v = y -> {u | [v / y]u e. C} = C)
7		oprabval4g.1	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
8		ax-17 925	. . . . . 6 |- ((w e. A /\ v e. B) -> A.x(w e. A /\ v e. B))
9		hbs1 986	. . . . . . . 8 |- ([w / x]f e. {u | [v / y]u e. C} -> A.x[w / x]f e. {u | [v / y]u e. C})
10	9	hbab 1096	. . . . . . 7 |- (z e. {f | [w / x]f e. {u | [v / y]u e. C}} -> A.x z e. {f | [w / x]f e. {u | [v / y]u e. C}})
11	10	hbeleq 1173	. . . . . 6 |- (z = {f | [w / x]f e. {u | [v / y]u e. C}} -> A.x z = {f | [w / x]f e. {u | [v / y]u e. C}})
12	8, 11	hban 704	. . . . 5 |- (((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}}) -> A.x((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}}))
13		ax-17 925	. . . . . 6 |- ((w e. A /\ v e. B) -> A.y(w e. A /\ v e. B))
14		hbs1 986	. . . . . . . . . 10 |- ([v / y]u e. C -> A.y[v / y]u e. C)
15	14	hbab 1096	. . . . . . . . 9 |- (f e. {u | [v / y]u e. C} -> A.y f e. {u | [v / y]u e. C})
16	15	hbsb 987	. . . . . . . 8 |- ([w / x]f e. {u | [v / y]u e. C} -> A.y[w / x]f e. {u | [v / y]u e. C})
17	16	hbab 1096	. . . . . . 7 |- (z e. {f | [w / x]f e. {u | [v / y]u e. C}} -> A.y z e. {f | [w / x]f e. {u | [v / y]u e. C}})
18	17	hbeleq 1173	. . . . . 6 |- (z = {f | [w / x]f e. {u | [v / y]u e. C}} -> A.y z = {f | [w / x]f e. {u | [v / y]u e. C}})
19	13, 18	hban 704	. . . . 5 |- (((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}}) -> A.y((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}}))
20		ax-17 925	. . . . 5 |- (((x e. A /\ y e. B) /\ z = C) -> A.w((x e. A /\ y e. B) /\ z = C))
21		ax-17 925	. . . . 5 |- (((x e. A /\ y e. B) /\ z = C) -> A.v((x e. A /\ y e. B) /\ z = C))
22		eleq1 1149	. . . . . . 7 |- (w = x -> (w e. A <-> x e. A))
23		eleq1 1149	. . . . . . 7 |- (v = y -> (v e. B <-> y e. B))
24	22, 23	bi2anan9 478	. . . . . 6 |- ((w = x /\ v = y) -> ((w e. A /\ v e. B) <-> (x e. A /\ y e. B)))
25	3, 6	sylan9eq 1144	. . . . . . 7 |- ((w = x /\ v = y) -> {f | [w / x]f e. {u | [v / y]u e. C}} = C)
26	25	cleq2d 1112	. . . . . 6 |- ((w = x /\ v = y) -> (z = {f | [w / x]f e. {u | [v / y]u e. C}} <-> z = C))
27	24, 26	anbi12d 476	. . . . 5 |- ((w = x /\ v = y) -> (((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}}) <-> ((x e. A /\ y e. B) /\ z = C)))
28	12, 19, 20, 21, 27	cbvoprab12 3028	. . . 4 |- {<.<.w, v>., z>. | ((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}})} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
29	7, 28	eqtr4 1122	. . 3 |- F = {<.<.w, v>., z>. | ((w e. A /\ v e. B) /\ z = {f | [w / x]f e. {u | [v / y]u e. C}})}
30	3, 6, 29	oprabval2g 3050	. 2 |- ((x e. A /\ y e. B /\ C e. V) -> (xFy) = C)
31		elisset 1354	. 2 |- (C e. D -> C e. V)
32	30, 31	syl3an3 621	1 |- ((x e. A /\ y e. B /\ C e. D) -> (xFy) = C)

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581   = weq 797  [wsb 852  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  elrnoprab 3054  mapxpen 3390  ruclem13 4897

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004

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