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Theorem satefvfmla0 33679
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 33675 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2 incom 4149 . . . . . . . . 9 ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3 sqxpexg 7668 . . . . . . . . . 10 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 inex1g 5264 . . . . . . . . . 10 ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
53, 4syl 17 . . . . . . . . 9 (𝑀𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
62, 5eqeltrid 2841 . . . . . . . 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 33617 . . . . . 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 6828 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12 satfun 33672 . . . . . . . 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 6656 . . . . . 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 6507 . . . . . 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 7804 . . . . . 6 ∅ ∈ ω
1918a1i 11 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
2018a1i 11 . . . . . . . . 9 (𝑀𝑉 → ∅ ∈ ω)
21 satfdmfmla 33661 . . . . . . . . 9 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
226, 20, 21mpd3an23 1462 . . . . . . . 8 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
2322eqcomd 2742 . . . . . . 7 (𝑀𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
2423eleq2d 2822 . . . . . 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 7856 . . . . 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 1370 . . . 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 33674 . . . . 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 1370 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))})
36 elmapi 8709 . . . . . . . . 9 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
37 simpl 483 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38 fmla0xp 33644 . . . . . . . . . . . . . . . 16 (Fmla‘∅) = ({∅} × (ω × ω))
3938eleq2i 2828 . . . . . . . . . . . . . . 15 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40 elxp 5644 . . . . . . . . . . . . . . 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 7932 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (1st𝑦) ∈ ω)
4342ad2antll 726 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st𝑦) ∈ ω)
44 vex 3445 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
45 vex 3445 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
4644, 45op2ndd 7911 . . . . . . . . . . . . . . . . . . 19 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd𝑋) = 𝑦)
4746fveq2d 6830 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (1st ‘(2nd𝑋)) = (1st𝑦))
4847eleq1d 2821 . . . . . . . . . . . . . . . . 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 1934 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5241, 51sylbi 216 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (1st ‘(2nd𝑋)) ∈ ω)
5352ad2antll 726 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (1st ‘(2nd𝑋)) ∈ ω)
5437, 53ffvelcdmd 7019 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀)
55 xp2nd 7933 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (2nd𝑦) ∈ ω)
5655ad2antll 726 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd𝑦) ∈ ω)
5746fveq2d 6830 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘(2nd𝑋)) = (2nd𝑦))
5857eleq1d 2821 . . . . . . . . . . . . . . . . 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 1934 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6241, 61sylbi 216 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (2nd ‘(2nd𝑋)) ∈ ω)
6362ad2antll 726 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (2nd ‘(2nd𝑋)) ∈ ω)
6437, 63ffvelcdmd 7019 . . . . . . . . . . 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 5697 . . . . . . . 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 6839 . . . . . . 7 (𝑎‘(2nd ‘(2nd𝑋))) ∈ V
7372epeli 5527 . . . . . 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 3410 . . . 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 1540  wex 1780  wcel 2105  {crab 3403  Vcvv 3441  cin 3897  c0 4270  𝒫 cpw 4548  {csn 4574  cop 4580   ciun 4942   class class class wbr 5093   E cep 5524   × cxp 5619  dom cdm 5621  Fun wfun 6474  wf 6476  cfv 6480  (class class class)co 7338  ωcom 7781  1st c1st 7898  2nd c2nd 7899  m cmap 8687   Sat csat 33597  Fmlacfmla 33598   Sat csate 33599
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-inf2 9499  ax-ac2 10321
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-se 5577  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-isom 6489  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-om 7782  df-1st 7900  df-2nd 7901  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-2o 8369  df-er 8570  df-map 8689  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-card 9797  df-ac 9974  df-goel 33601  df-gona 33602  df-goal 33603  df-sat 33604  df-sate 33605  df-fmla 33606
This theorem is referenced by:  sategoelfvb  33680  prv1n  33692
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