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Theorem satfvel 35397
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 35396 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
2 ffvelcdm 7101 . . . . 5 ((((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω) ∧ 𝑈 ∈ (Fmla‘ω)) → (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω))
3 fvex 6920 . . . . . . 7 (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ V
43elpw 4609 . . . . . 6 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀m ω) ↔ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω))
5 ssel 3989 . . . . . . 7 ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆 ∈ (𝑀m ω)))
6 elmapi 8888 . . . . . . 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 2106  wss 3963  𝒫 cpw 4605  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  m cmap 8865   Sat csat 35321  Fmlacfmla 35322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-ac 10154  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-fmla 35330
This theorem is referenced by:  satef  35401  prv0  35415
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