Theorem goel 35352

Description: A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership v_i ∈ v_j is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)

Assertion

Ref	Expression
goel	⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)

Proof of Theorem goel

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7434	. 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘⟨𝐼, 𝐽⟩)
2		df-goel 35345	. . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
3	2	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩))
4		opeq2 4874	. . . 4 ⊢ (𝑥 = ⟨𝐼, 𝐽⟩ → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
5	4	adantl 481	. . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = ⟨𝐼, 𝐽⟩) → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
6		opelxpi 5722	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
7		opex 5469	. . . 4 ⊢ ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V
8	7	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V)
9	3, 5, 6, 8	fvmptd 7023	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘⟨𝐼, 𝐽⟩) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
10	1, 9	eqtrid 2789	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683  ‘cfv 6561  (class class class)co 7431  ωcom 7887  ∈_𝑔cgoe 35338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-goel 35345

This theorem is referenced by:  goelel3xp  35353  goeleq12bg  35354  sat1el2xp  35384  fmla0xp  35388  fmlaomn0  35395  gonan0  35397  goaln0  35398  gonar  35400  goalr  35402  fmla0disjsuc  35403  satfv0fvfmla0  35418  sategoelfvb  35424  prv1n  35436