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Theorem satefvfmla0 34012
Description: The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.)
Assertion
Ref Expression
satefvfmla0 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
Distinct variable groups:   𝑀,𝑎   𝑉,𝑎   𝑋,𝑎

Proof of Theorem satefvfmla0
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satefv 34008 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2 incom 4161 . . . . . . . . 9 ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3 sqxpexg 7689 . . . . . . . . . 10 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 inex1g 5276 . . . . . . . . . 10 ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
53, 4syl 17 . . . . . . . . 9 (𝑀𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
62, 5eqeltrid 2842 . . . . . . . 8 (𝑀𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
76ancli 549 . . . . . . 7 (𝑀𝑉 → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
87adantr 481 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9 satom 33950 . . . . . 6 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
108, 9syl 17 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
1110fveq1d 6844 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12 satfun 34005 . . . . . . . 8 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
138, 12syl 17 . . . . . . 7 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
1413ffund 6672 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1510eqcomd 2742 . . . . . . 7 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1615funeqd 6523 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ↔ Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)))
1714, 16mpbird 256 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
18 peano1 7825 . . . . . 6 ∅ ∈ ω
1918a1i 11 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
2018a1i 11 . . . . . . . . 9 (𝑀𝑉 → ∅ ∈ ω)
21 satfdmfmla 33994 . . . . . . . . 9 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
226, 20, 21mpd3an23 1463 . . . . . . . 8 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
2322eqcomd 2742 . . . . . . 7 (𝑀𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
2423eleq2d 2823 . . . . . 6 (𝑀𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
2524biimpa 477 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26 eqid 2736 . . . . . 6 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
2726fviunfun 7877 . . . . 5 ((Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ∧ ∅ ∈ ω ∧ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)) → ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
2817, 19, 25, 27syl3anc 1371 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
2911, 28eqtrd 2776 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30 simpl 483 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑀𝑉)
316adantr 481 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32 simpr 485 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33 eqid 2736 . . . . . 6 (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
3433satfv0fvfmla0 34007 . . . . 5 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))})
3530, 31, 32, 34syl3anc 1371 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))})
36 elmapi 8787 . . . . . . . . 9 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
37 simpl 483 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38 fmla0xp 33977 . . . . . . . . . . . . . . . 16 (Fmla‘∅) = ({∅} × (ω × ω))
3938eleq2i 2829 . . . . . . . . . . . . . . 15 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40 elxp 5656 . . . . . . . . . . . . . . 15 (𝑋 ∈ ({∅} × (ω × ω)) ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
4139, 40bitri 274 . . . . . . . . . . . . . 14 (𝑋 ∈ (Fmla‘∅) ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
42 xp1st 7953 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (1st𝑦) ∈ ω)
4342ad2antll 727 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st𝑦) ∈ ω)
44 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
45 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
4644, 45op2ndd 7932 . . . . . . . . . . . . . . . . . . 19 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd𝑋) = 𝑦)
4746fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (1st ‘(2nd𝑋)) = (1st𝑦))
4847eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨𝑥, 𝑦⟩ → ((1st ‘(2nd𝑋)) ∈ ω ↔ (1st𝑦) ∈ ω))
4948adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((1st ‘(2nd𝑋)) ∈ ω ↔ (1st𝑦) ∈ ω))
5043, 49mpbird 256 . . . . . . . . . . . . . . 15 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5150exlimivv 1935 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5241, 51sylbi 216 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (1st ‘(2nd𝑋)) ∈ ω)
5352ad2antll 727 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (1st ‘(2nd𝑋)) ∈ ω)
5437, 53ffvelcdmd 7036 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀)
55 xp2nd 7954 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (2nd𝑦) ∈ ω)
5655ad2antll 727 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd𝑦) ∈ ω)
5746fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘(2nd𝑋)) = (2nd𝑦))
5857eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(2nd𝑋)) ∈ ω ↔ (2nd𝑦) ∈ ω))
5958adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((2nd ‘(2nd𝑋)) ∈ ω ↔ (2nd𝑦) ∈ ω))
6056, 59mpbird 256 . . . . . . . . . . . . . . 15 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6160exlimivv 1935 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6241, 61sylbi 216 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (2nd ‘(2nd𝑋)) ∈ ω)
6362ad2antll 727 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (2nd ‘(2nd𝑋)) ∈ ω)
6437, 63ffvelcdmd 7036 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀)
6554, 64jca 512 . . . . . . . . . 10 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀))
6665ex 413 . . . . . . . . 9 (𝑎:ω⟶𝑀 → ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀)))
6736, 66syl 17 . . . . . . . 8 (𝑎 ∈ (𝑀m ω) → ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀)))
6867impcom 408 . . . . . . 7 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀))
69 brinxp 5710 . . . . . . . 8 (((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀) → ((𝑎‘(1st ‘(2nd𝑋))) E (𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))))
7069bicomd 222 . . . . . . 7 (((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀) → ((𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) E (𝑎‘(2nd ‘(2nd𝑋)))))
7168, 70syl 17 . . . . . 6 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) E (𝑎‘(2nd ‘(2nd𝑋)))))
72 fvex 6855 . . . . . . 7 (𝑎‘(2nd ‘(2nd𝑋))) ∈ V
7372epeli 5539 . . . . . 6 ((𝑎‘(1st ‘(2nd𝑋))) E (𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋))))
7471, 73bitrdi 286 . . . . 5 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))))
7574rabbidva 3414 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
7635, 75eqtrd 2776 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
7729, 76eqtrd 2776 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
781, 77eqtrd 2776 1 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  {crab 3407  Vcvv 3445  cin 3909  c0 4282  𝒫 cpw 4560  {csn 4586  cop 4592   ciun 4954   class class class wbr 5105   E cep 5536   × cxp 5631  dom cdm 5633  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  ωcom 7802  1st c1st 7919  2nd c2nd 7920  m cmap 8765   Sat csat 33930  Fmlacfmla 33931   Sat csate 33932
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-ac 10052  df-goel 33934  df-gona 33935  df-goal 33936  df-sat 33937  df-sate 33938  df-fmla 33939
This theorem is referenced by:  sategoelfvb  34013  prv1n  34025
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