Theorem sate0 36002

Description: The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.)

Assertion

Ref	Expression
sate0	⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))

Proof of Theorem sate0

Step	Hyp	Ref	Expression
1		0ex 5268	. . 3 ⊢ ∅ ∈ V
2		satefv 36001	. . 3 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ 𝑉) → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
3	1, 2	mpan 703	. 2 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4		xp0 5759	. . . . . . 7 ⊢ (∅ × ∅) = ∅
5	4	ineq2i 4166	. . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6		in0 4348	. . . . . 6 ⊢ ( E ∩ ∅) = ∅
7	5, 6	eqtri 2785	. . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅
8	7	oveq2i 7428	. . . 4 ⊢ (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
9	8	fveq1i 6883	. . 3 ⊢ ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
10	9	fveq1i 6883	. 2 ⊢ (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
11	3, 10	eqtrdi 2813	1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∩ cin 3901  ∅c0 4282   E cep 5558   × cxp 5657  ‘cfv 6537  (class class class)co 7417  ωcom 7866   Sat csat 35923   Sat_∈ csate 35925

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-sate 35931

This theorem is used by:  prv0  36017