Theorem satefv 35943

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 35873	. . 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 5695	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀))
5	4	ineq2d 4173	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀)))
6	3, 5	oveq12d 7437	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀))))
7	6	fveq1d 6887	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
8	7	adantr 486	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
9		simpr 490	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
10	8, 9	fveq12d 6892	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
11	10	adantl 487	. 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 3478	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
13	12	adantr 486	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑀 ∈ V)
14		elex 3478	. . 3 ⊢ (𝑈 ∈ 𝑊 → 𝑈 ∈ V)
15	14	adantl 487	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑈 ∈ V)
16		fvexd 6900	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V)
17	2, 11, 13, 15, 16	ovmpod 7571	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905   E cep 5562   × cxp 5661  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  ωcom 7868   Sat csat 35865   Sat_∈ csate 35867

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-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-sate 35873

This theorem is used by:  sate0  35944  satef  35945  satefvfmla0  35947  satefvfmla1  35954