Theorem satfvel 35439

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 35438	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))
2		ffvelcdm 7076	. . . . 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 6894	. . . . . . 7 ⊢ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ V
4	3	elpw 4584	. . . . . 6 ⊢ ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ∈ 𝒫 (𝑀 ↑m ω) ↔ (((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀 ↑m ω))
5		ssel 3957	. . . . . . 7 ⊢ ((((𝑀 Sat 𝐸)‘ω)‘𝑈) ⊆ (𝑀 ↑m ω) → (𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈) → 𝑆 ∈ (𝑀 ↑m ω)))
6		elmapi 8868	. . . . . . 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 217	. . . . 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 412	. . 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 1110	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   ⊆ wss 3931  𝒫 cpw 4580  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ωcom 7866   ↑_m cmap 8845   Sat csat 35363  Fmlacfmla 35364

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-ac2 10482

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9958  df-ac 10135  df-goel 35367  df-gona 35368  df-goal 35369  df-sat 35370  df-fmla 35372

This theorem is referenced by:  satef  35443  prv0  35457