Theorem goel 35529

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 7370	. 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘⟨𝐼, 𝐽⟩)
2		df-goel 35522	. . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
3	2	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩))
4		opeq2 4817	. . . 4 ⊢ (𝑥 = ⟨𝐼, 𝐽⟩ → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
5	4	adantl 481	. . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = ⟨𝐼, 𝐽⟩) → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
6		opelxpi 5668	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
7		opex 5416	. . . 4 ⊢ ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V
8	7	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V)
9	3, 5, 6, 8	fvmptd 6955	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘⟨𝐼, 𝐽⟩) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
10	1, 9	eqtrid 2783	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  ⟨cop 4573   ↦ cmpt 5166   × cxp 5629  ‘cfv 6498  (class class class)co 7367  ωcom 7817  ∈_𝑔cgoe 35515

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-goel 35522

This theorem is referenced by:  goelel3xp  35530  goeleq12bg  35531  sat1el2xp  35561  fmla0xp  35565  fmlaomn0  35572  gonan0  35574  goaln0  35575  gonar  35577  goalr  35579  fmla0disjsuc  35580  satfv0fvfmla0  35595  sategoelfvb  35601  prv1n  35613