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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmlafv | Structured version Visualization version GIF version |
Description: The valid Godel formulas of height 𝑁 is the domain of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 15-Sep-2023.) |
Ref | Expression |
---|---|
fmlafv | ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fmla 35305 | . . 3 ⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ suc ω → Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))) |
3 | fveq2 6919 | . . . 4 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁)) | |
4 | 3 | dmeqd 5929 | . . 3 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁)) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝑁 ∈ suc ω ∧ 𝑛 = 𝑁) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁)) |
6 | id 22 | . 2 ⊢ (𝑁 ∈ suc ω → 𝑁 ∈ suc ω) | |
7 | fvex 6932 | . . . 4 ⊢ ((∅ Sat ∅)‘𝑁) ∈ V | |
8 | 7 | dmex 7945 | . . 3 ⊢ dom ((∅ Sat ∅)‘𝑁) ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V) |
10 | 2, 5, 6, 9 | fvmptd 7034 | 1 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 Vcvv 3482 ∅c0 4347 ↦ cmpt 5252 dom cdm 5699 suc csuc 6396 ‘cfv 6572 (class class class)co 7445 ωcom 7899 Sat csat 35296 Fmlacfmla 35297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-iota 6524 df-fun 6574 df-fv 6580 df-fmla 35305 |
This theorem is referenced by: fmla 35341 fmla0 35342 fmlasuc0 35344 satfdmfmla 35360 |
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