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Theorem sate0 32787
 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 5175 . . 3 ∅ ∈ V
2 satefv 32786 . . 3 ((∅ ∈ V ∧ 𝑈𝑉) → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
31, 2mpan 689 . 2 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4 xp0 5982 . . . . . . 7 (∅ × ∅) = ∅
54ineq2i 4136 . . . . . 6 ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6 in0 4299 . . . . . 6 ( E ∩ ∅) = ∅
75, 6eqtri 2821 . . . . 5 ( E ∩ (∅ × ∅)) = ∅
87oveq2i 7146 . . . 4 (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
98fveq1i 6646 . . 3 ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
109fveq1i 6646 . 2 (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
113, 10eqtrdi 2849 1 (𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  ∅c0 4243   E cep 5429   × cxp 5517  ‘cfv 6324  (class class class)co 7135  ωcom 7562   Sat csat 32708   Sat∈ csate 32710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-sate 32716 This theorem is referenced by:  prv0  32802
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