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Theorem sate0 35399
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 5312 . . 3 ∅ ∈ V
2 satefv 35398 . . 3 ((∅ ∈ V ∧ 𝑈𝑉) → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
31, 2mpan 690 . 2 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4 xp0 6179 . . . . . . 7 (∅ × ∅) = ∅
54ineq2i 4224 . . . . . 6 ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6 in0 4400 . . . . . 6 ( E ∩ ∅) = ∅
75, 6eqtri 2762 . . . . 5 ( E ∩ (∅ × ∅)) = ∅
87oveq2i 7441 . . . 4 (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
98fveq1i 6907 . . 3 ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
109fveq1i 6907 . 2 (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
113, 10eqtrdi 2790 1 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  c0 4338   E cep 5587   × cxp 5686  cfv 6562  (class class class)co 7430  ωcom 7886   Sat csat 35320   Sat csate 35322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-sate 35328
This theorem is referenced by:  prv0  35414
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