Theorem satfdmfmla 32668

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 5204	. . . . . . 7 ⊢ ∅ ∈ V
2	1, 1	pm3.2i 473	. . . . . 6 ⊢ (∅ ∈ V ∧ ∅ ∈ V)
3	2	jctr 527	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
4	3	3adant3 1127	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (∅ ∈ V ∧ ∅ ∈ V)))
5		satfdm 32637	. . . 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 6663	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑛) = ((𝑀 Sat 𝐸)‘𝑁))
8	7	dmeqd 5767	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑀 Sat 𝐸)‘𝑁))
9		fveq2 6663	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((∅ Sat ∅)‘𝑛) = ((∅ Sat ∅)‘𝑁))
10	9	dmeqd 5767	. . . . . 6 ⊢ (𝑛 = 𝑁 → dom ((∅ Sat ∅)‘𝑛) = dom ((∅ Sat ∅)‘𝑁))
11	8, 10	eqeq12d 2836	. . . . 5 ⊢ (𝑛 = 𝑁 → (dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) ↔ dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
12	11	rspcv 3615	. . . 4 ⊢ (𝑁 ∈ ω → (∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((∅ Sat ∅)‘𝑛) → dom ((𝑀 Sat 𝐸)‘𝑁) = dom ((∅ Sat ∅)‘𝑁)))
13	12	3ad2ant3 1130	. . 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 6256	. . . 4 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc ω)
16	15	3ad2ant3 1130	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → 𝑁 ∈ suc ω)
17		fmlafv 32648	. . 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 2858	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  ∀wral 3137  Vcvv 3491  ∅c0 4284  dom cdm 5548  suc csuc 6186  ‘cfv 6348  (class class class)co 7149  ωcom 7573   Sat csat 32604  Fmlacfmla 32605

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-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-inf2 9097

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  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-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  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-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-goel 32608  df-goal 32610  df-sat 32611  df-fmla 32613

This theorem is referenced by:  satffunlem1lem2  32671  satffunlem2lem2  32674  satff  32678  satefvfmla0  32686  satefvfmla1  32693