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 ∈ A ∧ y ∈ B) ∧ z = C)}

Assertion

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

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   = weq 797  [wsb 852  {cab 1090   = wceq 1091   ∈ 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

metamath.org