Theorem satfvel 32772

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‘1_o), etc.) so that 𝑈 is true under the assignment 𝑆. (Contributed by AV, 29-Oct-2023.)

Assertion

Ref	Expression
satfvel	⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)

Proof of Theorem satfvel

Step	Hyp	Ref	Expression
1		satfun 32771	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))
2		ffvelrn 6826	. . . . 5 ⊢ ((((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω) ∧ 𝑈 ∈ (Fmla‘ω)) → (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀 ↑m ω))
3		fvex 6658	. . . . . . 7 ⊢ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ V
4	3	elpw 4501	. . . . . 6 ⊢ ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀 ↑m ω) ↔ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀 ↑m ω))
5		ssel 3908	. . . . . . 7 ⊢ ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀 ↑m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆 ∈ (𝑀 ↑m ω)))
6		elmapi 8411	. . . . . . 7 ⊢ (𝑆 ∈ (𝑀 ↑m ω) → 𝑆:ω⟶𝑀)
7	5, 6	syl6 35	. . . . . 6 ⊢ ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀 ↑m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀))
8	4, 7	sylbi 220	. . . . 5 ⊢ ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀 ↑m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀))
9	2, 8	syl 17	. . . 4 ⊢ ((((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω) ∧ 𝑈 ∈ (Fmla‘ω)) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀))
10	9	ex 416	. . 3 ⊢ (((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω) → (𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀)))
11	1, 10	syl 17	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆:ω⟶𝑀)))
12	11	3imp 1108	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ⊆ wss 3881  𝒫 cpw 4497  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ωcom 7560   ↑_m cmap 8389   Sat csat 32696  Fmlacfmla 32697

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-card 9352  df-ac 9527  df-goel 32700  df-gona 32701  df-goal 32702  df-sat 32703  df-fmla 32705

This theorem is referenced by:  satef  32776  prv0  32790