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Theorem satfvel 32733
Description: An element of the value of the satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at the code 𝑈 for a wff using ∈ , ⊼ , ∀ is a valuation 𝑆:ω⟶𝑀 of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1o), etc.) so that 𝑈 is true under the assignment 𝑆. (Contributed by AV, 29-Oct-2023.)
Assertion
Ref Expression
satfvel (((𝑀𝑉𝐸𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)

Proof of Theorem satfvel
StepHypRef Expression
1 satfun 32732 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
2 ffvelrn 6831 . . . . 5 ((((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω) ∧ 𝑈 ∈ (Fmla‘ω)) → (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω))
3 fvex 6665 . . . . . . 7 (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ V
43elpw 4515 . . . . . 6 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω) ↔ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω))
5 ssel 3935 . . . . . . 7 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆 ∈ (𝑀m ω)))
6 elmapi 8415 . . . . . . 7 (𝑆 ∈ (𝑀m ω) → 𝑆:ω⟶𝑀)
75, 6syl6 35 . . . . . 6 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀))
84, 7sylbi 220 . . . . 5 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀))
92, 8syl 17 . . . 4 ((((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω) ∧ 𝑈 ∈ (Fmla‘ω)) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀))
109ex 416 . . 3 (((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω) → (𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀)))
111, 10syl 17 . 2 ((𝑀𝑉𝐸𝑊) → (𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀)))
12113imp 1108 1 (((𝑀𝑉𝐸𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2114  wss 3908  𝒫 cpw 4511  wf 6330  cfv 6334  (class class class)co 7140  ωcom 7565  m cmap 8393   Sat csat 32657  Fmlacfmla 32658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-ac2 9874
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-card 9356  df-ac 9531  df-goel 32661  df-gona 32662  df-goal 32663  df-sat 32664  df-fmla 32666
This theorem is referenced by:  satef  32737  prv0  32751
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