Theorem satfdmfmla 35594

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 5252	. . . . . . 7 ⊢ ∅ ∈ V
2	1, 1	pm3.2i 470	. . . . . 6 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
3	2	jctr 524	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
4	3	3adant3 1132	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5		satfdm 35563	. . . 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 6834	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
8	7	dmeqd 5854	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9		fveq2 6834	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
10	9	dmeqd 5854	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
11	8, 10	eqeq12d 2752	. . . . 5 ⊢ (𝑛 = 𝑁 → (dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) ↔ dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
12	11	rspcv 3572	. . . 4 ⊢ (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13	12	3ad2ant3 1135	. . 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 6392	. . . 4 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16	15	3ad2ant3 1135	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17		fmlafv 35574	. . 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 2774	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440  ∅c0 4285  dom cdm 5624  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808   Sat csat 35530  Fmlacfmla 35531

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-goel 35534  df-goal 35536  df-sat 35537  df-fmla 35539

This theorem is referenced by:  satffunlem1lem2  35597  satffunlem2lem2  35600  satff  35604  satefvfmla0  35612  satefvfmla1  35619