Theorem goel 32594

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 7159	. 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘⟨𝐼, 𝐽⟩)
2		df-goel 32587	. . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
3	2	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩))
4		opeq2 4804	. . . 4 ⊢ (𝑥 = ⟨𝐼, 𝐽⟩ → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
5	4	adantl 484	. . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = ⟨𝐼, 𝐽⟩) → ⟨∅, 𝑥⟩ = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
6		opelxpi 5592	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨𝐼, 𝐽⟩ ∈ (ω × ω))
7		opex 5356	. . . 4 ⊢ ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V
8	7	a1i 11	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ⟨∅, ⟨𝐼, 𝐽⟩⟩ ∈ V)
9	3, 5, 6, 8	fvmptd 6775	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘⟨𝐼, 𝐽⟩) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
10	1, 9	syl5eq 2868	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  ⟨cop 4573   ↦ cmpt 5146   × cxp 5553  ‘cfv 6355  (class class class)co 7156  ωcom 7580  ∈_𝑔cgoe 32580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-goel 32587

This theorem is referenced by:  goelel3xp  32595  goeleq12bg  32596  sat1el2xp  32626  fmla0xp  32630  fmlaomn0  32637  gonan0  32639  goaln0  32640  gonar  32642  goalr  32644  fmla0disjsuc  32645  satfv0fvfmla0  32660  sategoelfvb  32666  prv1n  32678