Theorem oprabex3 3046

Description: Existence of an operation abstraction (special case).

Hypotheses

Ref	Expression
oprabex3.1	⊢ H ∈ V
oprabex3.2	⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}

Assertion

Ref	Expression
oprabex3	⊢ F ∈ V

Distinct variable group(s):   x,y,z,w,v,u,f,H   x,R,y,z

Proof of Theorem oprabex3

Step	Hyp	Ref	Expression
1		oprabex3.1	. . 3 ⊢ H ∈ V
2	1, 1	xpex 2488	. 2 ⊢ (H × H) ∈ V
3		moeq 1431	. . . . . 6 ⊢ ∃*z z = R
4	3	mosubop 1911	. . . . 5 ⊢ ∃*z∃u∃f(y = ⟨u, f⟩ ∧ z = R)
5	4	mosubop 1911	. . . 4 ⊢ ∃*z∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R))
6		anass 336	. . . . . . . 8 ⊢ (((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ (x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)))
7	6	bi2ex 734	. . . . . . 7 ⊢ (∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃u∃f(x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)))
8		19.42vv 968	. . . . . . 7 ⊢ (∃u∃f(x = ⟨w, v⟩ ∧ (y = ⟨u, f⟩ ∧ z = R)) ↔ (x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
9	7, 8	bitr 151	. . . . . 6 ⊢ (∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ (x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
10	9	bi2ex 734	. . . . 5 ⊢ (∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
11	10	bimo 1031	. . . 4 ⊢ (∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R) ↔ ∃*z∃w∃v(x = ⟨w, v⟩ ∧ ∃u∃f(y = ⟨u, f⟩ ∧ z = R)))
12	5, 11	mpbir 165	. . 3 ⊢ ∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R)
13	12	a1i 7	. 2 ⊢ ((x ∈ (H × H) ∧ y ∈ (H × H)) → ∃*z∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))
14		oprabex3.2	. 2 ⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ (H × H) ∧ y ∈ (H × H)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = R))}
15	2, 2, 13, 14	oprabex 3044	1 ⊢ F ∈ V

Syntax hints:   ∧ wa 196  ∃wex 678  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   × cxp 2408  {copab2 3002

This theorem is referenced by:  axaddex 4059  axmulex 4060

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-fn 2433  df-oprab 3004

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