Theorem prv 34419

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 7418	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 Sat∈ 𝑢) = (𝑀 Sat∈ 𝑈))
2		simpl 484	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑚 = 𝑀)
3	2	oveq1d 7424	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 ↑m ω) = (𝑀 ↑m ω))
4	1, 3	eqeq12d 2749	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω) ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))
5		df-prv 34337	. 2 ⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
6	4, 5	brabga 5535	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  (class class class)co 7409  ωcom 7855   ↑_m cmap 8820   Sat_∈ csate 34329  ⊧cprv 34330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-ov 7412  df-prv 34337

This theorem is referenced by:  elnanelprv  34420  prv0  34421  prv1n  34422