Theorem sate0 35253

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 5302	. . 3 ⊢ ∅ ∈ V
2		satefv 35252	. . 3 ⊢ ((∅ ∈ V ∧ 𝑈 ∈ 𝑉) → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
3	1, 2	mpan 688	. 2 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈))
4		xp0 6159	. . . . . . 7 ⊢ (∅ × ∅) = ∅
5	4	ineq2i 4207	. . . . . 6 ⊢ ( E ∩ (∅ × ∅)) = ( E ∩ ∅)
6		in0 4389	. . . . . 6 ⊢ ( E ∩ ∅) = ∅
7	5, 6	eqtri 2754	. . . . 5 ⊢ ( E ∩ (∅ × ∅)) = ∅
8	7	oveq2i 7424	. . . 4 ⊢ (∅ Sat ( E ∩ (∅ × ∅))) = (∅ Sat ∅)
9	8	fveq1i 6891	. . 3 ⊢ ((∅ Sat ( E ∩ (∅ × ∅)))‘ω) = ((∅ Sat ∅)‘ω)
10	9	fveq1i 6891	. 2 ⊢ (((∅ Sat ( E ∩ (∅ × ∅)))‘ω)‘𝑈) = (((∅ Sat ∅)‘ω)‘𝑈)
11	3, 10	eqtrdi 2782	1 ⊢ (𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3462   ∩ cin 3945  ∅c0 4322   E cep 5575   × cxp 5670  ‘cfv 6543  (class class class)co 7413  ωcom 7865   Sat csat 35174   Sat_∈ csate 35176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-sate 35182

This theorem is referenced by:  prv0  35268