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Theorem sate0 35420
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 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))

Proof of Theorem sate0
StepHypRef Expression
1 0ex 5307 . . 3 ∅ ∈ V
2 satefv 35419 . . 3 ((∅ ∈ V ∧ 𝑈𝑉) → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
31, 2mpan 690 . 2 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4 xp0 6178 . . . . . . 7 (∅ × ∅) = ∅
54ineq2i 4217 . . . . . 6 ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6 in0 4395 . . . . . 6 ( E ∩ ∅) = ∅
75, 6eqtri 2765 . . . . 5 ( E ∩ (∅ × ∅)) = ∅
87oveq2i 7442 . . . 4 (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
98fveq1i 6907 . . 3 ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
109fveq1i 6907 . 2 (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
113, 10eqtrdi 2793 1 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  c0 4333   E cep 5583   × cxp 5683  cfv 6561  (class class class)co 7431  ωcom 7887   Sat csat 35341   Sat csate 35343
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sate 35349
This theorem is referenced by:  prv0  35435
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