Theorem oprabval4g 4031

Description: Value of an operation given by an ordered-pair class abstraction. (This is the operation analog of fvopab2 3791.)

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 1583	. . . . 5 |- (x = w -> {u | [v / y]u e. C} = {f | [w / x]f e. {u | [v / y]u e. C}})
2	1	eqcomd 1480	. . . 4 |- (x = w -> {f | [w / x]f e. {u | [v / y]u e. C}} = {u | [v / y]u e. C})
3	2	equcoms 1130	. . 3 |- (w = x -> {f | [w / x]f e. {u | [v / y]u e. C}} = {u | [v / y]u e. C})
4		sbab 1583	. . . . 5 |- (y = v -> C = {u | [v / y]u e. C})
5	4	eqcomd 1480	. . . 4 |- (y = v -> {u | [v / y]u e. C} = C)
6	5	equcoms 1130	. . 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-17 971	. . . . . 6 |- ((w e. A /\ v e. B) -> A.x(w e. A /\ v e. B))
9		hbs1 1332	. . . . . . . 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 1467	. . . . . . 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 1567	. . . . . 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 1009	. . . . 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-17 971	. . . . . 6 |- ((w e. A /\ v e. B) -> A.y(w e. A /\ v e. B))
14		hbs1 1332	. . . . . . . . . 10 |- ([v / y]u e. C -> A.y[v / y]u e. C)
15	14	hbab 1467	. . . . . . . . 9 |- (f e. {u | [v / y]u e. C} -> A.y f e. {u | [v / y]u e. C})
16	15	hbsb 1333	. . . . . . . 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 1467	. . . . . . 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 1567	. . . . . 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 1009	. . . . 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-17 971	. . . . 5 |- (((x e. A /\ y e. B) /\ z = C) -> A.w((x e. A /\ y e. B) /\ z = C))
21		ax-17 971	. . . . 5 |- (((x e. A /\ y e. B) /\ z = C) -> A.v((x e. A /\ y e. B) /\ z = C))
22		eleq1 1534	. . . . . . 7 |- (w = x -> (w e. A <-> x e. A))
23		eleq1 1534	. . . . . . 7 |- (v = y -> (v e. B <-> y e. B))
24	22, 23	bi2anan9 632	. . . . . 6 |- ((w = x /\ v = y) -> ((w e. A /\ v e. B) <-> (x e. A /\ y e. B)))
25	3, 6	sylan9eq 1527	. . . . . . 7 |- ((w = x /\ v = y) -> {f | [w / x]f e. {u | [v / y]u e. C}} = C)
26	25	eqeq2d 1486	. . . . . 6 |- ((w = x /\ v = y) -> (z = {f | [w / x]f e. {u | [v / y]u e. C}} <-> z = C))
27	24, 26	anbi12d 628	. . . . 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 3998	. . . 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 1498	. . 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 4027	. 2 |- ((x e. A /\ y e. B /\ C e. V) -> (xFy) = C)
31		elisset 1817	. 2 |- (C e. D -> C e. V)
32	30, 31	syl3an3 861	1 |- ((x e. A /\ y e. B /\ C e. D) -> (xFy) = C)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  (class class class)co 3963  {copab2 3964

This theorem is referenced by:  elrnoprabg 4124  mapxpen 4495

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  df-oprab 3966

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