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Theorem oprabex 7672
Description: Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
Hypotheses
Ref Expression
oprabex.1 𝐴 ∈ V
oprabex.2 𝐵 ∈ V
oprabex.3 ((𝑥𝐴𝑦𝐵) → ∃*𝑧𝜑)
oprabex.4 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
Assertion
Ref Expression
oprabex 𝐹 ∈ V
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem oprabex
StepHypRef Expression
1 oprabex.4 . 2 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
2 oprabex.3 . . . . 5 ((𝑥𝐴𝑦𝐵) → ∃*𝑧𝜑)
3 moanimv 2707 . . . . 5 (∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → ∃*𝑧𝜑))
42, 3mpbir 234 . . . 4 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝜑)
54funoprab 7267 . . 3 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)}
6 oprabex.1 . . . . 5 𝐴 ∈ V
7 oprabex.2 . . . . 5 𝐵 ∈ V
86, 7xpex 7470 . . . 4 (𝐴 × 𝐵) ∈ V
9 dmoprabss 7249 . . . 4 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
108, 9ssexi 5212 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ∈ V
11 funex 6973 . . 3 ((Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ∧ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ∈ V) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ∈ V)
125, 10, 11mp2an 691 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ∈ V
131, 12eqeltri 2912 1 𝐹 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  ∃*wmo 2622  Vcvv 3480   × cxp 5540  dom cdm 5542  Fun wfun 6337  {coprab 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-oprab 7153
This theorem is referenced by:  oprabex3  7673  joinfval  17611  meetfval  17625
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