Theorem satefv 35401

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 35331	. . 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 22	. . . . . . 7 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
4	3	sqxpeqd 5670	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀))
5	4	ineq2d 4183	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀)))
6	3, 5	oveq12d 7405	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀))))
7	6	fveq1d 6860	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
8	7	adantr 480	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
9		simpr 484	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
10	8, 9	fveq12d 6865	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
11	10	adantl 481	. 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 3468	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
13	12	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑀 ∈ V)
14		elex 3468	. . 3 ⊢ (𝑈 ∈ 𝑊 → 𝑈 ∈ V)
15	14	adantl 481	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑈 ∈ V)
16		fvexd 6873	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V)
17	2, 11, 13, 15, 16	ovmpod 7541	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913   E cep 5537   × cxp 5636  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ωcom 7842   Sat csat 35323   Sat_∈ csate 35325

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-sate 35331

This theorem is referenced by:  sate0  35402  satef  35403  satefvfmla0  35405  satefvfmla1  35412