Theorem prv 35941

Description: The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀 ↑_m ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by AV, 5-Nov-2023.)

Assertion

Ref	Expression
prv	⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))

Proof of Theorem prv

Dummy variables 𝑚 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 7432	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 Sat∈ 𝑢) = (𝑀 Sat∈ 𝑈))
2		simpl 488	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑚 = 𝑀)
3	2	oveq1d 7438	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 ↑m ω) = (𝑀 ↑m ω))
4	1, 3	eqeq12d 2782	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω) ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))
5		df-prv 35859	. 2 ⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
6	4, 5	brabga 5523	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   class class class wbr 5114  (class class class)co 7423  ωcom 7871   ↑_m cmap 8833   Sat_∈ csate 35851  ⊧cprv 35852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-iota 6499  df-fv 6551  df-ov 7426  df-prv 35859

This theorem is used by:  elnanelprv  35942  prv0  35943  prv1n  35944