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Theorem sate0 35609
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 5252 . . 3 ∅ ∈ V
2 satefv 35608 . . 3 ((∅ ∈ V ∧ 𝑈𝑉) → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
31, 2mpan 690 . 2 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4 xp0 5724 . . . . . . 7 (∅ × ∅) = ∅
54ineq2i 4169 . . . . . 6 ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6 in0 4347 . . . . . 6 ( E ∩ ∅) = ∅
75, 6eqtri 2759 . . . . 5 ( E ∩ (∅ × ∅)) = ∅
87oveq2i 7369 . . . 4 (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
98fveq1i 6835 . . 3 ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
109fveq1i 6835 . 2 (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
113, 10eqtrdi 2787 1 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2113  Vcvv 3440  cin 3900  c0 4285   E cep 5523   × cxp 5622  cfv 6492  (class class class)co 7358  ωcom 7808   Sat csat 35530   Sat csate 35532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-sate 35538
This theorem is referenced by:  prv0  35624
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