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Theorem satefvfmla1 34019
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 7391 . . . . 5 𝑋 ∈ V
32jctr 525 . . . 4 (𝑀𝑉 → (𝑀𝑉𝑋 ∈ V))
433ad2ant1 1133 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉𝑋 ∈ V))
5 satefv 34008 . . 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 7689 . . . . . . . 8 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
8 inex2g 5277 . . . . . . . 8 ((𝑀 × 𝑀) ∈ V → ( E ∩ (𝑀 × 𝑀)) ∈ V)
97, 8syl 17 . . . . . . 7 (𝑀𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
109ancli 549 . . . . . 6 (𝑀𝑉 → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
11103ad2ant1 1133 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
12 satom 33950 . . . . 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 6844 . . 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 34005 . . . . . . 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 6672 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1813eqcomd 2742 . . . . . 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 6523 . . . . 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 256 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
21 1onn 8586 . . . . 5 1o ∈ ω
2221a1i 11 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 1o ∈ ω)
2312goelgoanfmla1 34018 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
24233adant1 1130 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
2521a1i 11 . . . . . . 7 (𝑀𝑉 → 1o ∈ ω)
26 satfdmfmla 33994 . . . . . . 7 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ 1o ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
279, 25, 26mpd3an23 1463 . . . . . 6 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
28273ad2ant1 1133 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
2924, 28eleqtrrd 2841 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o))
30 eqid 2736 . . . . 5 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
3130fviunfun 7877 . . . 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 1371 . . 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 2776 . 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 34017 . . . 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 1163 . . 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 5157 . . . . . . 7 ((𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ ((𝑎𝐼) E (𝑎𝐽) ∧ (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽)))
37 elmapi 8787 . . . . . . . . . . . . . . 15 (𝑎 ∈ (𝑀m ω) → 𝑎:ω⟶𝑀)
38 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐼 ∈ ω) → (𝑎𝐼) ∈ 𝑀)
3938ex 413 . . . . . . . . . . . . . . 15 (𝑎:ω⟶𝑀 → (𝐼 ∈ ω → (𝑎𝐼) ∈ 𝑀))
4037, 39syl 17 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑀m ω) → (𝐼 ∈ ω → (𝑎𝐼) ∈ 𝑀))
41 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐽 ∈ ω) → (𝑎𝐽) ∈ 𝑀)
4241ex 413 . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . 11 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑎 ∈ (𝑀m ω) → ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀)))
4746imp 407 . . . . . . . . . 10 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀))
48 brxp 5681 . . . . . . . . . 10 ((𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽) ↔ ((𝑎𝐼) ∈ 𝑀 ∧ (𝑎𝐽) ∈ 𝑀))
4947, 48sylibr 233 . . . . . . . . 9 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽))
5049biantrud 532 . . . . . . . 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 6855 . . . . . . . . 9 (𝑎𝐽) ∈ V
5251epeli 5539 . . . . . . . 8 ((𝑎𝐼) E (𝑎𝐽) ↔ (𝑎𝐼) ∈ (𝑎𝐽))
5350, 52bitr3di 285 . . . . . . 7 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (((𝑎𝐼) E (𝑎𝐽) ∧ (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽)) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
5436, 53bitrid 282 . . . . . 6 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ (𝑎𝐼) ∈ (𝑎𝐽)))
5554notbid 317 . . . . 5 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (¬ (𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ ¬ (𝑎𝐼) ∈ (𝑎𝐽)))
56 brin 5157 . . . . . . 7 ((𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿) ↔ ((𝑎𝐾) E (𝑎𝐿) ∧ (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿)))
57 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐾 ∈ ω) → (𝑎𝐾) ∈ 𝑀)
5857ex 413 . . . . . . . . . . . . . . 15 (𝑎:ω⟶𝑀 → (𝐾 ∈ ω → (𝑎𝐾) ∈ 𝑀))
5937, 58syl 17 . . . . . . . . . . . . . 14 (𝑎 ∈ (𝑀m ω) → (𝐾 ∈ ω → (𝑎𝐾) ∈ 𝑀))
60 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝑎:ω⟶𝑀𝐿 ∈ ω) → (𝑎𝐿) ∈ 𝑀)
6160ex 413 . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . 11 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑎 ∈ (𝑀m ω) → ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀)))
6665imp 407 . . . . . . . . . 10 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀))
67 brxp 5681 . . . . . . . . . 10 ((𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿) ↔ ((𝑎𝐾) ∈ 𝑀 ∧ (𝑎𝐿) ∈ 𝑀))
6866, 67sylibr 233 . . . . . . . . 9 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿))
6968biantrud 532 . . . . . . . 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 6855 . . . . . . . . 9 (𝑎𝐿) ∈ V
7170epeli 5539 . . . . . . . 8 ((𝑎𝐾) E (𝑎𝐿) ↔ (𝑎𝐾) ∈ (𝑎𝐿))
7269, 71bitr3di 285 . . . . . . 7 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (((𝑎𝐾) E (𝑎𝐿) ∧ (𝑎𝐾)(𝑀 × 𝑀)(𝑎𝐿)) ↔ (𝑎𝐾) ∈ (𝑎𝐿)))
7356, 72bitrid 282 . . . . . 6 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → ((𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿) ↔ (𝑎𝐾) ∈ (𝑎𝐿)))
7473notbid 317 . . . . 5 (((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀m ω)) → (¬ (𝑎𝐾)( E ∩ (𝑀 × 𝑀))(𝑎𝐿) ↔ ¬ (𝑎𝐾) ∈ (𝑎𝐿)))
7555, 74orbi12d 917 . . . 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 3414 . . 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 2776 . 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 2780 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  cin 3909  𝒫 cpw 4560   ciun 4954   class class class wbr 5105   E cep 5536   × cxp 5631  dom cdm 5633  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  ωcom 7802  1oc1o 8405  m cmap 8765  𝑔cgoe 33927  𝑔cgna 33928   Sat csat 33930  Fmlacfmla 33931   Sat csate 33932
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-ac 10052  df-goel 33934  df-gona 33935  df-goal 33936  df-sat 33937  df-sate 33938  df-fmla 33939
This theorem is referenced by:  elnanelprv  34023
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