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Theorem satefvfmla0 35402
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 35398 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2 incom 4216 . . . . . . . . 9 ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3 sqxpexg 7773 . . . . . . . . . 10 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 inex1g 5324 . . . . . . . . . 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 548 . . . . . . 7 (𝑀𝑉 → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
87adantr 480 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9 satom 35340 . . . . . 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 6908 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12 satfun 35395 . . . . . . . 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 6740 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1510eqcomd 2740 . . . . . . 7 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1615funeqd 6589 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ↔ Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)))
1714, 16mpbird 257 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
18 peano1 7910 . . . . . 6 ∅ ∈ ω
1918a1i 11 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
2018a1i 11 . . . . . . . . 9 (𝑀𝑉 → ∅ ∈ ω)
21 satfdmfmla 35384 . . . . . . . . 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 2740 . . . . . . 7 (𝑀𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
2423eleq2d 2824 . . . . . 6 (𝑀𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
2524biimpa 476 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26 eqid 2734 . . . . . 6 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
2726fviunfun 7967 . . . . 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 2774 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30 simpl 482 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑀𝑉)
316adantr 480 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32 simpr 484 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33 eqid 2734 . . . . . 6 (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
3433satfv0fvfmla0 35397 . . . . 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 8887 . . . . . . . . 9 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
37 simpl 482 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38 fmla0xp 35367 . . . . . . . . . . . . . . . 16 (Fmla‘∅) = ({∅} × (ω × ω))
3938eleq2i 2830 . . . . . . . . . . . . . . 15 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40 elxp 5711 . . . . . . . . . . . . . . 15 (𝑋 ∈ ({∅} × (ω × ω)) ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
4139, 40bitri 275 . . . . . . . . . . . . . 14 (𝑋 ∈ (Fmla‘∅) ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
42 xp1st 8044 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (1st𝑦) ∈ ω)
4342ad2antll 729 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st𝑦) ∈ ω)
44 vex 3481 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
45 vex 3481 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
4644, 45op2ndd 8023 . . . . . . . . . . . . . . . . . . 19 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd𝑋) = 𝑦)
4746fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (1st ‘(2nd𝑋)) = (1st𝑦))
4847eleq1d 2823 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨𝑥, 𝑦⟩ → ((1st ‘(2nd𝑋)) ∈ ω ↔ (1st𝑦) ∈ ω))
4948adantr 480 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((1st ‘(2nd𝑋)) ∈ ω ↔ (1st𝑦) ∈ ω))
5043, 49mpbird 257 . . . . . . . . . . . . . . 15 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5150exlimivv 1929 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5241, 51sylbi 217 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (1st ‘(2nd𝑋)) ∈ ω)
5352ad2antll 729 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (1st ‘(2nd𝑋)) ∈ ω)
5437, 53ffvelcdmd 7104 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀)
55 xp2nd 8045 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (2nd𝑦) ∈ ω)
5655ad2antll 729 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd𝑦) ∈ ω)
5746fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘(2nd𝑋)) = (2nd𝑦))
5857eleq1d 2823 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(2nd𝑋)) ∈ ω ↔ (2nd𝑦) ∈ ω))
5958adantr 480 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((2nd ‘(2nd𝑋)) ∈ ω ↔ (2nd𝑦) ∈ ω))
6056, 59mpbird 257 . . . . . . . . . . . . . . 15 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6160exlimivv 1929 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6241, 61sylbi 217 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (2nd ‘(2nd𝑋)) ∈ ω)
6362ad2antll 729 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (2nd ‘(2nd𝑋)) ∈ ω)
6437, 63ffvelcdmd 7104 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀)
6554, 64jca 511 . . . . . . . . . 10 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀))
6665ex 412 . . . . . . . . 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 407 . . . . . . 7 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀))
69 brinxp 5766 . . . . . . . 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 223 . . . . . . 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 6919 . . . . . . 7 (𝑎‘(2nd ‘(2nd𝑋))) ∈ V
7372epeli 5590 . . . . . 6 ((𝑎‘(1st ‘(2nd𝑋))) E (𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋))))
7471, 73bitrdi 287 . . . . 5 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))))
7574rabbidva 3439 . . . 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 2774 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
7729, 76eqtrd 2774 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
781, 77eqtrd 2774 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 206  wa 395   = wceq 1536  wex 1775  wcel 2105  {crab 3432  Vcvv 3477  cin 3961  c0 4338  𝒫 cpw 4604  {csn 4630  cop 4636   ciun 4995   class class class wbr 5147   E cep 5587   × cxp 5686  dom cdm 5688  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  ωcom 7886  1st c1st 8010  2nd c2nd 8011  m cmap 8864   Sat csat 35320  Fmlacfmla 35321   Sat csate 35322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-ac2 10500
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-ac 10153  df-goel 35324  df-gona 35325  df-goal 35326  df-sat 35327  df-sate 35328  df-fmla 35329
This theorem is referenced by:  sategoelfvb  35403  prv1n  35415
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