Theorem oprabex3 6335

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 4871	. 2 ⊢ (𝐻 × 𝐻) ∈ V
3		moeq 2995	. . . . . 6 ⊢ ∃*𝑧 𝑧 = 𝑅
4	3	mosubop 4821	. . . . 5 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
5	4	mosubop 4821	. . . 4 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
6		anass 401	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
7	6	2exbii 1655	. . . . . . 7 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
8		19.42vv 1963	. . . . . . 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 184	. . . . . 6 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
10	9	2exbii 1655	. . . . 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 2119	. . . 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 146	. . 3 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
13	12	a1i 9	. 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 6334	1 ⊢ 𝐹 ∈ V

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541  ∃*wmo 2083   ∈ wcel 2205  Vcvv 2815  ⟨cop 3697   × cxp 4752  {coprab 6059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-oprab 6062

This theorem is referenced by:  addvalex  8175