Theorem prv 35433

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 7440	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 Sat∈ 𝑢) = (𝑀 Sat∈ 𝑈))
2		simpl 482	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑚 = 𝑀)
3	2	oveq1d 7446	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 ↑m ω) = (𝑀 ↑m ω))
4	1, 3	eqeq12d 2753	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω) ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))
5		df-prv 35351	. 2 ⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
6	4, 5	brabga 5539	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   class class class wbr 5143  (class class class)co 7431  ωcom 7887   ↑_m cmap 8866   Sat_∈ csate 35343  ⊧cprv 35344

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434  df-prv 35351

This theorem is referenced by:  elnanelprv  35434  prv0  35435  prv1n  35436