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Theorem satfvel 35418
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 35417 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
2 ffvelcdm 7100 . . . . 5 ((((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω) ∧ 𝑈 ∈ (Fmla‘ω)) → (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω))
3 fvex 6918 . . . . . . 7 (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ V
43elpw 4603 . . . . . 6 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω) ↔ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω))
5 ssel 3976 . . . . . . 7 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆 ∈ (𝑀m ω)))
6 elmapi 8890 . . . . . . 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 217 . . . . 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 412 . . 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 1110 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 395  w3a 1086  wcel 2107  wss 3950  𝒫 cpw 4599  wf 6556  cfv 6560  (class class class)co 7432  ωcom 7888  m cmap 8867   Sat csat 35342  Fmlacfmla 35343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-ac 10157  df-goel 35346  df-gona 35347  df-goal 35348  df-sat 35349  df-fmla 35351
This theorem is referenced by:  satef  35422  prv0  35436
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