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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 32611 | . . 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 6663 | . . . 4 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁)) | |
4 | 3 | dmeqd 5767 | . . 3 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁)) |
5 | 4 | adantl 484 | . 2 ⊢ ((𝑁 ∈ suc ω ∧ 𝑛 = 𝑁) → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁)) |
6 | id 22 | . 2 ⊢ (𝑁 ∈ suc ω → 𝑁 ∈ suc ω) | |
7 | fvex 6676 | . . . 4 ⊢ ((∅ Sat ∅)‘𝑁) ∈ V | |
8 | 7 | dmex 7609 | . . 3 ⊢ dom ((∅ Sat ∅)‘𝑁) ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝑁 ∈ suc ω → dom ((∅ Sat ∅)‘𝑁) ∈ V) |
10 | 2, 5, 6, 9 | fvmptd 6768 | 1 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 ↦ cmpt 5139 dom cdm 5548 suc csuc 6186 ‘cfv 6348 (class class class)co 7149 ωcom 7573 Sat csat 32602 Fmlacfmla 32603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-fmla 32611 |
This theorem is referenced by: fmla 32647 fmla0 32648 fmlasuc0 32650 satfdmfmla 32666 |
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