Theorem satefv 35804

Description: The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)

Assertion

Ref	Expression
satefv	⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))

Proof of Theorem satefv

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

Step	Hyp	Ref	Expression
1		df-sate 35734	. . 3 ⊢ Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
2	1	a1i 11	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢)))
3		id 23	. . . . . . 7 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
4	3	sqxpeqd 5694	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀))
5	4	ineq2d 4181	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀)))
6	3, 5	oveq12d 7429	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀))))
7	6	fveq1d 6884	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
8	7	adantr 485	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
9		simpr 489	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
10	8, 9	fveq12d 6889	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
11	10	adantl 486	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑢 = 𝑈)) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
12		elex 3484	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
13	12	adantr 485	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑀 ∈ V)
14		elex 3484	. . 3 ⊢ (𝑈 ∈ 𝑊 → 𝑈 ∈ V)
15	14	adantl 486	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑈 ∈ V)
16		fvexd 6897	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V)
17	2, 11, 13, 15, 16	ovmpod 7563	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   E cep 5561   × cxp 5660  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  ωcom 7861   Sat csat 35726   Sat_∈ csate 35728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-sate 35734

This theorem is referenced by:  sate0  35805  satef  35806  satefvfmla0  35808  satefvfmla1  35815