Theorem oprabex3 7918

Description: Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)

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 groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓,𝐻   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝑅(𝑤,𝑣,𝑢,𝑓)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓)

Proof of Theorem oprabex3

Step	Hyp	Ref	Expression
1		oprabex3.1	. . 3 ⊢ 𝐻 ∈ V
2	1, 1	xpex 7695	. 2 ⊢ (𝐻 × 𝐻) ∈ V
3		moeq 3662	. . . . . 6 ⊢ ∃*𝑧 𝑧 = 𝑅
4	3	mosubop 5456	. . . . 5 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
5	4	mosubop 5456	. . . 4 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
6		anass 468	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
7	6	2exbii 1850	. . . . . . 7 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
8		19.42vv 1958	. . . . . . 7 ⊢ (∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
9	7, 8	bitri 275	. . . . . 6 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
10	9	2exbii 1850	. . . . 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	mobii 2545	. . . 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 231	. . 3 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
13	12	a1i 11	. 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 7917	1 ⊢ 𝐹 ∈ V

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2535  Vcvv 3437  ⟨cop 4583   × cxp 5619  {coprab 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-oprab 7359

This theorem is referenced by: (None)