Theorem goel 36081

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 7415	. 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘⟨𝐼, 𝐽⟩)
2		df-goel 36074	. . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
3	2	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩))
4		opeq2 4834	. . . 4 ⊢ (𝑥 = ⟨𝐼, 𝐽⟩ → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
5	4	adantl 487	. . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = ⟨𝐼, 𝐽⟩) → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
6		opelxpi 5688	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
7		opex 5432	. . . 4 ⊢ ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V
8	7	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V)
9	3, 5, 6, 8	fvmptd 6993	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘⟨𝐼, 𝐽⟩) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
10	1, 9	eqtrid 2808	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  ⟨cop 4590   ↦ cmpt 5186   × cxp 5649  ‘cfv 6531  (class class class)co 7412  ωcom 7866  ∈_𝑔cgoe 36067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-goel 36074

This theorem is used by:  goelel3xp  36082  goeleq12bg  36083  sat1el2xp  36113  fmla0xp  36117  fmlaomn0  36124  gonan0  36126  goaln0  36127  gonar  36129  goalr  36131  fmla0disjsuc  36132  satfv0fvfmla0  36147  sategoelfvb  36153  prv1n  36165