Theorem oprabval4g 4026

Description: Value of an operation given by an ordered-pair class abstraction. (This is the operation analog of fvopab2 3786.)

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 groups:   x,y,z,A   x,B,y,z   z,C

Proof of Theorem oprabval4g

Step	Hyp	Ref	Expression
1		sbab 1581	. . . . 5 |- (x = w -> {u | [v / y]u e. C} = {f | [w / x]f e. {u | [v / y]u e. C}})
2	1	eqcomd 1478	. . . 4 |- (x = w -> {f | [w / x]f e. {u | [v / y]u e. C}} = {u | [v / y]u e. C})
3	2	equcoms 1129	. . 3 |- (w = x -> {f | [w / x]f e. {u | [v / y]u e. C}} = {u | [v / y]u e. C})
4		sbab 1581	. . . . 5 |- (y = v -> C = {u | [v / y]u e. C})
5	4	eqcomd 1478	. . . 4 |- (y = v -> {u | [v / y]u e. C} = C)
6	5	equcoms 1129	. . 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-17 970	. . . . . 6 |- ((w e. A /\ v e. B) -> A.x(w e. A /\ v e. B))
9		hbs1 1331	. . . . . . . 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 1466	. . . . . . 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 1565	. . . . . 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 1008	. . . . 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-17 970	. . . . . 6 |- ((w e. A /\ v e. B) -> A.y(w e. A /\ v e. B))
14		hbs1 1331	. . . . . . . . . 10 |- ([v / y]u e. C -> A.y[v / y]u e. C)
15	14	hbab 1466	. . . . . . . . 9 |- (f e. {u | [v / y]u e. C} -> A.y f e. {u | [v / y]u e. C})
16	15	hbsb 1332	. . . . . . . 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 1466	. . . . . . 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 1565	. . . . . 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 1008	. . . . 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-17 970	. . . . 5 |- (((x e. A /\ y e. B) /\ z = C) -> A.w((x e. A /\ y e. B) /\ z = C))
21		ax-17 970	. . . . 5 |- (((x e. A /\ y e. B) /\ z = C) -> A.v((x e. A /\ y e. B) /\ z = C))
22		eleq1 1532	. . . . . . 7 |- (w = x -> (w e. A <-> x e. A))
23		eleq1 1532	. . . . . . 7 |- (v = y -> (v e. B <-> y e. B))
24	22, 23	bi2anan9 631	. . . . . 6 |- ((w = x /\ v = y) -> ((w e. A /\ v e. B) <-> (x e. A /\ y e. B)))
25	3, 6	sylan9eq 1525	. . . . . . 7 |- ((w = x /\ v = y) -> {f | [w / x]f e. {u | [v / y]u e. C}} = C)
26	25	eqeq2d 1484	. . . . . 6 |- ((w = x /\ v = y) -> (z = {f | [w / x]f e. {u | [v / y]u e. C}} <-> z = C))
27	24, 26	anbi12d 627	. . . . 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 3993	. . . 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 1496	. . 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 4022	. 2 |- ((x e. A /\ y e. B /\ C e. V) -> (xFy) = C)
31		elisset 1814	. 2 |- (C e. D -> C e. V)
32	30, 31	syl3an3 860	1 |- ((x e. A /\ y e. B /\ C e. D) -> (xFy) = C)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  [wsbc 1169  {cab 1462  Vcvv 1808  (class class class)co 3958  {copab2 3959

This theorem is referenced by:  elrnoprabg 4117  mapxpen 4484

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194  df-opr 3960  df-oprab 3961

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