Theorem prv 35663

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 7372	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 Sat∈ 𝑢) = (𝑀 Sat∈ 𝑈))
2		simpl 483	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑚 = 𝑀)
3	2	oveq1d 7378	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (𝑚 ↑m ω) = (𝑀 ↑m ω))
4	1, 3	eqeq12d 2756	. 2 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω) ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))
5		df-prv 35581	. 2 ⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
6	4, 5	brabga 5483	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   class class class wbr 5079  (class class class)co 7363  ωcom 7813   ↑_m cmap 8770   Sat_∈ csate 35573  ⊧cprv 35574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-iota 6448  df-fv 6500  df-ov 7366  df-prv 35581

This theorem is referenced by:  elnanelprv  35664  prv0  35665  prv1n  35666