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Theorem satefvfmla1 35410
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 1132 . . 3 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉𝑋 ∈ V))
5 satefv 35399 . . 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 7774 . . . . . . . 8 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
8 inex2g 5326 . . . . . . . 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 1132 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
12 satom 35341 . . . . 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 6909 . . 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 35396 . . . . . . 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 6741 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
1813eqcomd 2741 . . . . . 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 6590 . . . . 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 8677 . . . . 5 1o ∈ ω
2221a1i 11 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 1o ∈ ω)
2312goelgoanfmla1 35409 . . . . . 6 (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
24233adant1 1129 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
2521a1i 11 . . . . . . 7 (𝑀𝑉 → 1o ∈ ω)
26 satfdmfmla 35385 . . . . . . 7 ((𝑀𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ 1o ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
279, 25, 26mpd3an23 1462 . . . . . 6 (𝑀𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
28273ad2ant1 1132 . . . . 5 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o) = (Fmla‘1o))
2924, 28eleqtrrd 2842 . . . 4 ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1o))
30 eqid 2735 . . . . 5 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
3130fviunfun 7968 . . . 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 1370 . . 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 2775 . 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 35408 . . . 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 1162 . . 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 5200 . . . . . . 7 ((𝑎𝐼)( E ∩ (𝑀 × 𝑀))(𝑎𝐽) ↔ ((𝑎𝐼) E (𝑎𝐽) ∧ (𝑎𝐼)(𝑀 × 𝑀)(𝑎𝐽)))
37 elmapi 8888 . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . 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 5738 . . . . . . . . . 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 6920 . . . . . . . . 9 (𝑎𝐽) ∈ V
5251epeli 5591 . . . . . . . 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 5200 . . . . . . 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 1134 . . . . . . . . . . 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 5738 . . . . . . . . . 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 6920 . . . . . . . . 9 (𝑎𝐿) ∈ V
7170epeli 5591 . . . . . . . 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 918 . . . 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 3440 . . 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 2775 . 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 2779 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 847  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cin 3962  𝒫 cpw 4605   ciun 4996   class class class wbr 5148   E cep 5588   × cxp 5687  dom cdm 5689  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  m cmap 8865  𝑔cgoe 35318  𝑔cgna 35319   Sat csat 35321  Fmlacfmla 35322   Sat csate 35323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-ac 10154  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-sate 35329  df-fmla 35330
This theorem is referenced by:  elnanelprv  35414
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