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Theorem goel 35332
Description: A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership vi vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
Assertion
Ref Expression
goel ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)

Proof of Theorem goel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . 2 (𝐼𝑔𝐽) = (∈𝑔‘⟨𝐼, 𝐽⟩)
2 df-goel 35325 . . . 4 𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
32a1i 11 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩))
4 opeq2 4879 . . . 4 (𝑥 = ⟨𝐼, 𝐽⟩ → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
54adantl 481 . . 3 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = ⟨𝐼, 𝐽⟩) → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
6 opelxpi 5726 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
7 opex 5475 . . . 4 ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V
87a1i 11 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V)
93, 5, 6, 8fvmptd 7023 . 2 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘⟨𝐼, 𝐽⟩) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
101, 9eqtrid 2787 1 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cop 4637  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  ωcom 7887  𝑔cgoe 35318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-goel 35325
This theorem is referenced by:  goelel3xp  35333  goeleq12bg  35334  sat1el2xp  35364  fmla0xp  35368  fmlaomn0  35375  gonan0  35377  goaln0  35378  gonar  35380  goalr  35382  fmla0disjsuc  35383  satfv0fvfmla0  35398  sategoelfvb  35404  prv1n  35416
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