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| Mirrors > Home > MPE Home > Th. List > Mathboxes > goel | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| goel | ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ov 7403 | . 2 ⊢ (𝐼∈𝑔𝐽) = (∈𝑔‘〈𝐼, 𝐽〉) | |
| 2 | df-goel 35703 | . . . 4 ⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ 〈∅, 𝑥〉) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ 〈∅, 𝑥〉)) |
| 4 | opeq2 4835 | . . . 4 ⊢ (𝑥 = 〈𝐼, 𝐽〉 → 〈∅, 𝑥〉 = 〈∅, 〈𝐼, 𝐽〉〉) | |
| 5 | 4 | adantl 486 | . . 3 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ 𝑥 = 〈𝐼, 𝐽〉) → 〈∅, 𝑥〉 = 〈∅, 〈𝐼, 𝐽〉〉) |
| 6 | opelxpi 5689 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈𝐼, 𝐽〉 ∈ (ω × ω)) | |
| 7 | opex 5436 | . . . 4 ⊢ 〈∅, 〈𝐼, 𝐽〉〉 ∈ V | |
| 8 | 7 | a1i 11 | . . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → 〈∅, 〈𝐼, 𝐽〉〉 ∈ V) |
| 9 | 3, 5, 6, 8 | fvmptd 6987 | . 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∈𝑔‘〈𝐼, 𝐽〉) = 〈∅, 〈𝐼, 𝐽〉〉) |
| 10 | 1, 9 | eqtrid 2812 | 1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ∅c0 4288 〈cop 4591 ↦ cmpt 5186 × cxp 5650 ‘cfv 6525 (class class class)co 7400 ωcom 7850 ∈𝑔cgoe 35696 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-pr 5395 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-iota 6481 df-fun 6527 df-fv 6533 df-ov 7403 df-goel 35703 |
| This theorem is referenced by: goelel3xp 35711 goeleq12bg 35712 sat1el2xp 35742 fmla0xp 35746 fmlaomn0 35753 gonan0 35755 goaln0 35756 gonar 35758 goalr 35760 fmla0disjsuc 35761 satfv0fvfmla0 35776 sategoelfvb 35782 prv1n 35794 |
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