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Theorem satefvfmla0 35423
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 35419 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2 incom 4209 . . . . . . . . 9 ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3 sqxpexg 7775 . . . . . . . . . 10 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 inex1g 5319 . . . . . . . . . 10 ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
53, 4syl 17 . . . . . . . . 9 (𝑀𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
62, 5eqeltrid 2845 . . . . . . . 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 35361 . . . . . 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 35416 . . . . . . . 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 2743 . . . . . . 7 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1615funeqd 6588 . . . . . 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 35405 . . . . . . . . 9 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
226, 20, 21mpd3an23 1465 . . . . . . . 8 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
2322eqcomd 2743 . . . . . . 7 (𝑀𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
2423eleq2d 2827 . . . . . 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 2737 . . . . . 6 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
2726fviunfun 7969 . . . . 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 1373 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
2911, 28eqtrd 2777 . . 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 2737 . . . . . 6 (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
3433satfv0fvfmla0 35418 . . . . 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 1373 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))})
36 elmapi 8889 . . . . . . . . 9 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
37 simpl 482 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38 fmla0xp 35388 . . . . . . . . . . . . . . . 16 (Fmla‘∅) = ({∅} × (ω × ω))
3938eleq2i 2833 . . . . . . . . . . . . . . 15 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40 elxp 5708 . . . . . . . . . . . . . . 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 8046 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (1st𝑦) ∈ ω)
4342ad2antll 729 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st𝑦) ∈ ω)
44 vex 3484 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
45 vex 3484 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
4644, 45op2ndd 8025 . . . . . . . . . . . . . . . . . . 19 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd𝑋) = 𝑦)
4746fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (1st ‘(2nd𝑋)) = (1st𝑦))
4847eleq1d 2826 . . . . . . . . . . . . . . . . 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 1932 . . . . . . . . . . . . . 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 7105 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀)
55 xp2nd 8047 . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . 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 1932 . . . . . . . . . . . . . 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 7105 . . . . . . . . . . 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 5764 . . . . . . . 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 5586 . . . . . 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 3443 . . . 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 2777 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
7729, 76eqtrd 2777 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
781, 77eqtrd 2777 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 1540  wex 1779  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  c0 4333  𝒫 cpw 4600  {csn 4626  cop 4632   ciun 4991   class class class wbr 5143   E cep 5583   × cxp 5683  dom cdm 5685  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  1st c1st 8012  2nd c2nd 8013  m cmap 8866   Sat csat 35341  Fmlacfmla 35342   Sat csate 35343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-ac 10156  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-sate 35349  df-fmla 35350
This theorem is referenced by:  sategoelfvb  35424  prv1n  35436
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