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Theorem oprabex3 5784
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)) ∧ ∃𝑤𝑣𝑢𝑓((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
StepHypRef Expression
1 oprabex3.1 . . 3 𝐻 ∈ V
21, 1xpex 4481 . 2 (𝐻 × 𝐻) ∈ V
3 moeq 2739 . . . . . 6 ∃*𝑧 𝑧 = 𝑅
43mosubop 4434 . . . . 5 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
54mosubop 4434 . . . 4 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
6 anass 387 . . . . . . . 8 (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
762exbii 1513 . . . . . . 7 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
8 19.42vv 1804 . . . . . . 7 (∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
97, 8bitri 177 . . . . . 6 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
1092exbii 1513 . . . . 5 (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
1110mobii 1953 . . . 4 (∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
125, 11mpbir 138 . . 3 ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
1312a1i 9 . 2 ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))
14 oprabex3.2 . 2 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
152, 2, 13, 14oprabex 5783 1 𝐹 ∈ V
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  ∃*wmo 1917  Vcvv 2574  cop 3406   × cxp 4371  {coprab 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-oprab 5544
This theorem is referenced by:  addvalex  6978
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