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Theorem satefvfmla1 35430
Description: The simplified satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
Hypothesis
Ref Expression
satfv1fvfmla1.x 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
Assertion
Ref Expression
satefvfmla1 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))})
Distinct variable groups:   𝐼,𝑎   𝐽,𝑎   𝐾,𝑎   𝐿,𝑎   𝑀,𝑎   𝑉,𝑎
Allowed substitution hint:   𝑋(𝑎)

Proof of Theorem satefvfmla1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 satfv1fvfmla1.x . . . . . 6 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
21ovexi 7465 . . . . 5 𝑋 ∈ V
32jctr 524 . . . 4 (𝑀𝑉 → (𝑀𝑉𝑋 ∈ V))
433ad2ant1 1134 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉𝑋 ∈ V))
5 satefv 35419 . . 3 ((𝑀𝑉𝑋 ∈ V) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
64, 5syl 17 . 2 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
7 sqxpexg 7775 . . . . . . . 8 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
8 inex2g 5320 . . . . . . . 8 ((𝑀 × 𝑀) ∈ V → ( E ∩ (𝑀 × 𝑀)) ∈ V)
97, 8syl 17 . . . . . . 7 (𝑀𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
109ancli 548 . . . . . 6 (𝑀𝑉 → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
11103ad2ant1 1134 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
12 satom 35361 . . . . 5 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
1311, 12syl 17 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
1413fveq1d 6908 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
15 satfun 35416 . . . . . . 7 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
1611, 15syl 17 . . . . . 6 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
1716ffund 6740 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1813eqcomd 2743 . . . . . 6 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1918funeqd 6588 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ↔ Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)))
2017, 19mpbird 257 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
21 1onn 8678 . . . . 5 1o ∈ ω
2221a1i 11 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 1o ∈ ω)
2312goelgoanfmla1 35429 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
24233adant1 1131 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
2521a1i 11 . . . . . . 7 (𝑀𝑉 → 1o ∈ ω)
26 satfdmfmla 35405 . . . . . . 7 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ 1o ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
279, 25, 26mpd3an23 1465 . . . . . 6 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
28273ad2ant1 1134 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
2924, 28eleqtrrd 2844 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o))
30 eqid 2737 . . . . 5 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
3130fviunfun 7969 . . . 4 ((Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ∧ 1o ∈ ω ∧ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)) → ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)‘𝑋))
3220, 22, 29, 31syl3anc 1373 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ( 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)‘𝑋))
3314, 32eqtrd 2777 . 2 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)‘𝑋))
341satfv1fvfmla1 35428 . . . 4 (((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ∨ ¬ (𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿))})
3510, 34syl3an1 1164 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ∨ ¬ (𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿))})
36 brin 5195 . . . . . . 7 ((𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ ((𝑎𝐼) E (𝑎𝐽) ∧ (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽)))
37 elmapi 8889 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
38 ffvelcdm 7101 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐼 ∈ ω) → (𝑎𝐼) ∈ 𝑀)
3938ex 412 . . . . . . . . . . . . . . 15 (𝑎:ω⟶𝑀 → (𝐼 ∈ ω → (𝑎𝐼) ∈ 𝑀))
4037, 39syl 17 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑀m ω) → (𝐼 ∈ ω → (𝑎𝐼) ∈ 𝑀))
41 ffvelcdm 7101 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐽 ∈ ω) → (𝑎𝐽) ∈ 𝑀)
4241ex 412 . . . . . . . . . . . . . . 15 (𝑎:ω⟶𝑀 → (𝐽 ∈ ω → (𝑎𝐽) ∈ 𝑀))
4337, 42syl 17 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑀m ω) → (𝐽 ∈ ω → (𝑎𝐽) ∈ 𝑀))
4440, 43anim12d 609 . . . . . . . . . . . . 13 (𝑎 ∈ (𝑀m ω) → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀)))
4544com12 32 . . . . . . . . . . . 12 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎 ∈ (𝑀m ω) → ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀)))
46453ad2ant2 1135 . . . . . . . . . . 11 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑎 ∈ (𝑀m ω) → ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀)))
4746imp 406 . . . . . . . . . 10 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀))
48 brxp 5734 . . . . . . . . . 10 ((𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽) ↔ ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀))
4947, 48sylibr 234 . . . . . . . . 9 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽))
5049biantrud 531 . . . . . . . 8 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐼) E (𝑎𝐽) ↔ ((𝑎𝐼) E (𝑎𝐽) ∧ (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽))))
51 fvex 6919 . . . . . . . . 9 (𝑎𝐽) ∈ V
5251epeli 5586 . . . . . . . 8 ((𝑎𝐼) E (𝑎𝐽) ↔ (𝑎𝐼) ∈ (𝑎𝐽))
5350, 52bitr3di 286 . . . . . . 7 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (((𝑎𝐼) E (𝑎𝐽) ∧ (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽)) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
5436, 53bitrid 283 . . . . . 6 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
5554notbid 318 . . . . 5 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (¬ (𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ ¬ (𝑎𝐼) ∈ (𝑎𝐽)))
56 brin 5195 . . . . . . 7 ((𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿) ↔ ((𝑎𝐾) E (𝑎𝐿) ∧ (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿)))
57 ffvelcdm 7101 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐾 ∈ ω) → (𝑎𝐾) ∈ 𝑀)
5857ex 412 . . . . . . . . . . . . . . 15 (𝑎:ω⟶𝑀 → (𝐾 ∈ ω → (𝑎𝐾) ∈ 𝑀))
5937, 58syl 17 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑀m ω) → (𝐾 ∈ ω → (𝑎𝐾) ∈ 𝑀))
60 ffvelcdm 7101 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐿 ∈ ω) → (𝑎𝐿) ∈ 𝑀)
6160ex 412 . . . . . . . . . . . . . . 15 (𝑎:ω⟶𝑀 → (𝐿 ∈ ω → (𝑎𝐿) ∈ 𝑀))
6237, 61syl 17 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑀m ω) → (𝐿 ∈ ω → (𝑎𝐿) ∈ 𝑀))
6359, 62anim12d 609 . . . . . . . . . . . . 13 (𝑎 ∈ (𝑀m ω) → ((𝐾 ∈ ω ∧ 𝐿 ∈ ω) → ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀)))
6463com12 32 . . . . . . . . . . . 12 ((𝐾 ∈ ω ∧ 𝐿 ∈ ω) → (𝑎 ∈ (𝑀m ω) → ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀)))
65643ad2ant3 1136 . . . . . . . . . . 11 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑎 ∈ (𝑀m ω) → ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀)))
6665imp 406 . . . . . . . . . 10 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀))
67 brxp 5734 . . . . . . . . . 10 ((𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿) ↔ ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀))
6866, 67sylibr 234 . . . . . . . . 9 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿))
6968biantrud 531 . . . . . . . 8 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐾) E (𝑎𝐿) ↔ ((𝑎𝐾) E (𝑎𝐿) ∧ (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿))))
70 fvex 6919 . . . . . . . . 9 (𝑎𝐿) ∈ V
7170epeli 5586 . . . . . . . 8 ((𝑎𝐾) E (𝑎𝐿) ↔ (𝑎𝐾) ∈ (𝑎𝐿))
7269, 71bitr3di 286 . . . . . . 7 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (((𝑎𝐾) E (𝑎𝐿) ∧ (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿)) ↔ (𝑎𝐾) ∈ (𝑎𝐿)))
7356, 72bitrid 283 . . . . . 6 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿) ↔ (𝑎𝐾) ∈ (𝑎𝐿)))
7473notbid 318 . . . . 5 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (¬ (𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿) ↔ ¬ (𝑎𝐾) ∈ (𝑎𝐿)))
7555, 74orbi12d 919 . . . 4 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((¬ (𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ∨ ¬ (𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿)) ↔ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))))
7675rabbidva 3443 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ∨ ¬ (𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))})
7735, 76eqtrd 2777 . 2 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))})
786, 33, 773eqtrd 2781 1 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  𝒫 cpw 4600   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  1oc1o 8499  m cmap 8866  𝑔cgoe 35338  𝑔cgna 35339   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-ifp 1064  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:  elnanelprv  35434
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