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Theorem sate0 33090
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 5200 . . 3 ∅ ∈ V
2 satefv 33089 . . 3 ((∅ ∈ V ∧ 𝑈𝑉) → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
31, 2mpan 690 . 2 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4 xp0 6021 . . . . . . 7 (∅ × ∅) = ∅
54ineq2i 4124 . . . . . 6 ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6 in0 4306 . . . . . 6 ( E ∩ ∅) = ∅
75, 6eqtri 2765 . . . . 5 ( E ∩ (∅ × ∅)) = ∅
87oveq2i 7224 . . . 4 (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
98fveq1i 6718 . . 3 ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
109fveq1i 6718 . 2 (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
113, 10eqtrdi 2794 1 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  Vcvv 3408  cin 3865  c0 4237   E cep 5459   × cxp 5549  cfv 6380  (class class class)co 7213  ωcom 7644   Sat csat 33011   Sat csate 33013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-sate 33019
This theorem is referenced by:  prv0  33105
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