Theorem satfdmfmla 32933

Description: The domain of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is the set of valid Godel formulas of height 𝑁. (Contributed by AV, 13-Oct-2023.)

Assertion

Ref	Expression
satfdmfmla	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Proof of Theorem satfdmfmla

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5175	. . . . . . 7 ⊢ ∅ ∈ V
2	1, 1	pm3.2i 474	. . . . . 6 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
3	2	jctr 528	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
4	3	3adant3 1133	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5		satfdm 32902	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
6	4, 5	syl 17	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛))
7		fveq2 6674	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
8	7	dmeqd 5748	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9		fveq2 6674	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
10	9	dmeqd 5748	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
11	8, 10	eqeq12d 2754	. . . . 5 ⊢ (𝑛 = 𝑁 → (dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) ↔ dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
12	11	rspcv 3521	. . . 4 ⊢ (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13	12	3ad2ant3 1136	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
14	6, 13	mpd 15	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
15		elelsuc 6244	. . . 4 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16	15	3ad2ant3 1136	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17		fmlafv 32913	. . 3 ⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
18	16, 17	syl 17	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
19	14, 18	eqtr4d 2776	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053  Vcvv 3398  ∅c0 4211  dom cdm 5525  suc csuc 6174  ‘cfv 6339  (class class class)co 7170  ωcom 7599   Sat csat 32869  Fmlacfmla 32870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-goel 32873  df-goal 32875  df-sat 32876  df-fmla 32878

This theorem is referenced by:  satffunlem1lem2  32936  satffunlem2lem2  32939  satff  32943  satefvfmla0  32951  satefvfmla1  32958