Theorem sate0 36267

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	⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))

Proof of Theorem sate0

Step	Hyp	Ref	Expression
1		0ex 5260	. . 3 ⊢ ∅ ∈ V
2		satefv 36266	. . 3 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ 𝑉) → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
3	1, 2	mpan 703	. 2 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4		xp0 5750	. . . . . . 7 ⊢ (∅ × ∅) = ∅
5	4	ineq2i 4114	. . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6		in0 4344	. . . . . 6 ⊢ ( E ∩ ∅) = ∅
7	5, 6	eqtri 2783	. . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅
8	7	oveq2i 7432	. . . 4 ⊢ (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
9	8	fveq1i 6887	. . 3 ⊢ ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
10	9	fveq1i 6887	. 2 ⊢ (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
11	3, 10	eqtrdi 2811	1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∩ cin 4089  ∅c0 4278   E cep 5549   × cxp 5648  ‘cfv 6539  (class class class)co 7421  ωcom 7878   Sat csat 36188   Sat_∈ csate 36190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-sate 36196

This theorem is used by:  prv0  36282