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Theorem satefvfmla0 32779
 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 32775 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2 incom 4131 . . . . . . . . 9 ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3 sqxpexg 7461 . . . . . . . . . 10 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 inex1g 5190 . . . . . . . . . 10 ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
53, 4syl 17 . . . . . . . . 9 (𝑀𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
62, 5eqeltrid 2897 . . . . . . . 8 (𝑀𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
76ancli 552 . . . . . . 7 (𝑀𝑉 → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
87adantr 484 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9 satom 32717 . . . . . 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 6651 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12 satfun 32772 . . . . . . . 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 6495 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1510eqcomd 2807 . . . . . . 7 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1615funeqd 6350 . . . . . 6 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ↔ Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)))
1714, 16mpbird 260 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
18 peano1 7585 . . . . . 6 ∅ ∈ ω
1918a1i 11 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
2018a1i 11 . . . . . . . . 9 (𝑀𝑉 → ∅ ∈ ω)
21 satfdmfmla 32761 . . . . . . . . 9 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
226, 20, 21mpd3an23 1460 . . . . . . . 8 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
2322eqcomd 2807 . . . . . . 7 (𝑀𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
2423eleq2d 2878 . . . . . 6 (𝑀𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
2524biimpa 480 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26 eqid 2801 . . . . . 6 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
2726fviunfun 7632 . . . . 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 1368 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
2911, 28eqtrd 2836 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30 simpl 486 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑀𝑉)
316adantr 484 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32 simpr 488 . . . . 5 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33 eqid 2801 . . . . . 6 (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
3433satfv0fvfmla0 32774 . . . . 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 1368 . . . 4 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋)))})
36 elmapi 8415 . . . . . . . . 9 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
37 simpl 486 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38 fmla0xp 32744 . . . . . . . . . . . . . . . 16 (Fmla‘∅) = ({∅} × (ω × ω))
3938eleq2i 2884 . . . . . . . . . . . . . . 15 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40 elxp 5546 . . . . . . . . . . . . . . 15 (𝑋 ∈ ({∅} × (ω × ω)) ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
4139, 40bitri 278 . . . . . . . . . . . . . 14 (𝑋 ∈ (Fmla‘∅) ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
42 xp1st 7707 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (1st𝑦) ∈ ω)
4342ad2antll 728 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st𝑦) ∈ ω)
44 vex 3447 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
45 vex 3447 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
4644, 45op2ndd 7686 . . . . . . . . . . . . . . . . . . 19 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd𝑋) = 𝑦)
4746fveq2d 6653 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (1st ‘(2nd𝑋)) = (1st𝑦))
4847eleq1d 2877 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨𝑥, 𝑦⟩ → ((1st ‘(2nd𝑋)) ∈ ω ↔ (1st𝑦) ∈ ω))
4948adantr 484 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((1st ‘(2nd𝑋)) ∈ ω ↔ (1st𝑦) ∈ ω))
5043, 49mpbird 260 . . . . . . . . . . . . . . 15 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5150exlimivv 1933 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1st ‘(2nd𝑋)) ∈ ω)
5241, 51sylbi 220 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (1st ‘(2nd𝑋)) ∈ ω)
5352ad2antll 728 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (1st ‘(2nd𝑋)) ∈ ω)
5437, 53ffvelrnd 6833 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀)
55 xp2nd 7708 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ω × ω) → (2nd𝑦) ∈ ω)
5655ad2antll 728 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd𝑦) ∈ ω)
5746fveq2d 6653 . . . . . . . . . . . . . . . . . 18 (𝑋 = ⟨𝑥, 𝑦⟩ → (2nd ‘(2nd𝑋)) = (2nd𝑦))
5857eleq1d 2877 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨𝑥, 𝑦⟩ → ((2nd ‘(2nd𝑋)) ∈ ω ↔ (2nd𝑦) ∈ ω))
5958adantr 484 . . . . . . . . . . . . . . . 16 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((2nd ‘(2nd𝑋)) ∈ ω ↔ (2nd𝑦) ∈ ω))
6056, 59mpbird 260 . . . . . . . . . . . . . . 15 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6160exlimivv 1933 . . . . . . . . . . . . . 14 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2nd ‘(2nd𝑋)) ∈ ω)
6241, 61sylbi 220 . . . . . . . . . . . . 13 (𝑋 ∈ (Fmla‘∅) → (2nd ‘(2nd𝑋)) ∈ ω)
6362ad2antll 728 . . . . . . . . . . . 12 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (2nd ‘(2nd𝑋)) ∈ ω)
6437, 63ffvelrnd 6833 . . . . . . . . . . 11 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀)
6554, 64jca 515 . . . . . . . . . 10 ((𝑎:ω⟶𝑀 ∧ (𝑀𝑉𝑋 ∈ (Fmla‘∅))) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀))
6665ex 416 . . . . . . . . 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 411 . . . . . . 7 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋))) ∈ 𝑀 ∧ (𝑎‘(2nd ‘(2nd𝑋))) ∈ 𝑀))
69 brinxp 5598 . . . . . . . 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 226 . . . . . . 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 6662 . . . . . . 7 (𝑎‘(2nd ‘(2nd𝑋))) ∈ V
7372epeli 5436 . . . . . 6 ((𝑎‘(1st ‘(2nd𝑋))) E (𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋))))
7471, 73syl6bb 290 . . . . 5 (((𝑀𝑉𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎‘(1st ‘(2nd𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))))
7574rabbidva 3428 . . . 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 2836 . . 3 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
7729, 76eqtrd 2836 . 2 ((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
781, 77eqtrd 2836 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 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {crab 3113  Vcvv 3444   ∩ cin 3883  ∅c0 4246  𝒫 cpw 4500  {csn 4528  ⟨cop 4534  ∪ ciun 4884   class class class wbr 5033   E cep 5432   × cxp 5521  dom cdm 5523  Fun wfun 6322  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  ωcom 7564  1st c1st 7673  2nd c2nd 7674   ↑m cmap 8393   Sat csat 32697  Fmlacfmla 32698   Sat∈ csate 32699 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-ac2 9878 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-card 9356  df-ac 9531  df-goel 32701  df-gona 32702  df-goal 32703  df-sat 32704  df-sate 32705  df-fmla 32706 This theorem is referenced by:  sategoelfvb  32780  prv1n  32792
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