satefvfmla1 - Mathbox for Mario Carneiro

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Theorem satefvfmla1 34714

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 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿))

**Assertion**
Ref	Expression
satefvfmla1	⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat_∈ 𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))})

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.

Step	Hyp	Ref	Expression
1		satfv1fvfmla1.x	. . . . . 6 ⊢ 𝑋 = ((𝐼∈_𝑔𝐽)⊼_𝑔(𝐾∈_𝑔𝐿))
2	1	ovexi 7445	. . . . 5 ⊢ 𝑋 ∈ V
3	2	jctr 523	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ V))
4	3	3ad2ant1 1131	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ V))
5		satefv 34703	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ V) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
6	4, 5	syl 17	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
7		sqxpexg 7744	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
8		inex2g 5319	. . . . . . . 8 ⊢ ((𝑀 × 𝑀) ∈ V → ( E ∩ (𝑀 × 𝑀)) ∈ V)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
10	9	ancli 547	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
11	10	3ad2ant1 1131	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
12		satom 34645	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
13	11, 12	syl 17	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
14	13	fveq1d 6892	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (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 34700	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑_m ω))
16	11, 15	syl 17	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑_m ω))
17	16	ffund 6720	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
18	13	eqcomd 2736	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
19	18	funeqd 6569	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ↔ Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)))
20	17, 19	mpbird 256	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
21		1onn 8641	. . . . 5 ⊢ 1_o ∈ ω
22	21	a1i 11	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 1_o ∈ ω)
23	1	2goelgoanfmla1 34713	. . . . . 6 ⊢ (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1_o))
24	23	3adant1 1128	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1_o))
25	21	a1i 11	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → 1_o ∈ ω)
26		satfdmfmla 34689	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ 1_o ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o) = (Fmla‘1_o))
27	9, 25, 26	mpd3an23 1461	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o) = (Fmla‘1_o))
28	27	3ad2ant1 1131	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o) = (Fmla‘1_o))
29	24, 28	eleqtrrd 2834	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o))
30		eqid 2730	. . . . 5 ⊢ ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
31	30	fviunfun 7933	. . . 4 ⊢ ((Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ∧ 1_o ∈ ω ∧ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)) → (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)‘𝑋))
32	20, 22, 29, 31	syl3anc 1369	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)‘𝑋))
33	14, 32	eqtrd 2770	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)‘𝑋))
34	1	satfv1fvfmla1 34712	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿))})
35	10, 34	syl3an1 1161	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿))})
36		brin 5199	. . . . . . 7 ⊢ ((𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ↔ ((𝑎‘𝐼) E (𝑎‘𝐽) ∧ (𝑎‘𝐼)(𝑀 × 𝑀)(𝑎‘𝐽)))
37		elmapi 8845	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → 𝑎:ω⟶𝑀)
38		ffvelcdm 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎:ω⟶𝑀 ∧ 𝐼 ∈ ω) → (𝑎‘𝐼) ∈ 𝑀)
39	38	ex 411	. . . . . . . . . . . . . . 15 ⊢ (𝑎:ω⟶𝑀 → (𝐼 ∈ ω → (𝑎‘𝐼) ∈ 𝑀))
40	37, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → (𝐼 ∈ ω → (𝑎‘𝐼) ∈ 𝑀))
41		ffvelcdm 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎:ω⟶𝑀 ∧ 𝐽 ∈ ω) → (𝑎‘𝐽) ∈ 𝑀)
42	41	ex 411	. . . . . . . . . . . . . . 15 ⊢ (𝑎:ω⟶𝑀 → (𝐽 ∈ ω → (𝑎‘𝐽) ∈ 𝑀))
43	37, 42	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → (𝐽 ∈ ω → (𝑎‘𝐽) ∈ 𝑀))
44	40, 43	anim12d 607	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → ((𝑎‘𝐼) ∈ 𝑀 ∧ (𝑎‘𝐽) ∈ 𝑀)))
45	44	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝑎 ∈ (𝑀 ↑_m ω) → ((𝑎‘𝐼) ∈ 𝑀 ∧ (𝑎‘𝐽) ∈ 𝑀)))
46	45	3ad2ant2 1132	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑎 ∈ (𝑀 ↑_m ω) → ((𝑎‘𝐼) ∈ 𝑀 ∧ (𝑎‘𝐽) ∈ 𝑀)))
47	46	imp 405	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘𝐼) ∈ 𝑀 ∧ (𝑎‘𝐽) ∈ 𝑀))
48		brxp 5724	. . . . . . . . . 10 ⊢ ((𝑎‘𝐼)(𝑀 × 𝑀)(𝑎‘𝐽) ↔ ((𝑎‘𝐼) ∈ 𝑀 ∧ (𝑎‘𝐽) ∈ 𝑀))
49	47, 48	sylibr 233	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → (𝑎‘𝐼)(𝑀 × 𝑀)(𝑎‘𝐽))
50	49	biantrud 530	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘𝐼) E (𝑎‘𝐽) ↔ ((𝑎‘𝐼) E (𝑎‘𝐽) ∧ (𝑎‘𝐼)(𝑀 × 𝑀)(𝑎‘𝐽))))
51		fvex 6903	. . . . . . . . 9 ⊢ (𝑎‘𝐽) ∈ V
52	51	epeli 5581	. . . . . . . 8 ⊢ ((𝑎‘𝐼) E (𝑎‘𝐽) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽))
53	50, 52	bitr3di 285	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → (((𝑎‘𝐼) E (𝑎‘𝐽) ∧ (𝑎‘𝐼)(𝑀 × 𝑀)(𝑎‘𝐽)) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
54	36, 53	bitrid 282	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ↔ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
55	54	notbid 317	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → (¬ (𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ↔ ¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽)))
56		brin 5199	. . . . . . 7 ⊢ ((𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿) ↔ ((𝑎‘𝐾) E (𝑎‘𝐿) ∧ (𝑎‘𝐾)(𝑀 × 𝑀)(𝑎‘𝐿)))
57		ffvelcdm 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎:ω⟶𝑀 ∧ 𝐾 ∈ ω) → (𝑎‘𝐾) ∈ 𝑀)
58	57	ex 411	. . . . . . . . . . . . . . 15 ⊢ (𝑎:ω⟶𝑀 → (𝐾 ∈ ω → (𝑎‘𝐾) ∈ 𝑀))
59	37, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → (𝐾 ∈ ω → (𝑎‘𝐾) ∈ 𝑀))
60		ffvelcdm 7082	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎:ω⟶𝑀 ∧ 𝐿 ∈ ω) → (𝑎‘𝐿) ∈ 𝑀)
61	60	ex 411	. . . . . . . . . . . . . . 15 ⊢ (𝑎:ω⟶𝑀 → (𝐿 ∈ ω → (𝑎‘𝐿) ∈ 𝑀))
62	37, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → (𝐿 ∈ ω → (𝑎‘𝐿) ∈ 𝑀))
63	59, 62	anim12d 607	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → ((𝐾 ∈ ω ∧ 𝐿 ∈ ω) → ((𝑎‘𝐾) ∈ 𝑀 ∧ (𝑎‘𝐿) ∈ 𝑀)))
64	63	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ω ∧ 𝐿 ∈ ω) → (𝑎 ∈ (𝑀 ↑_m ω) → ((𝑎‘𝐾) ∈ 𝑀 ∧ (𝑎‘𝐿) ∈ 𝑀)))
65	64	3ad2ant3 1133	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑎 ∈ (𝑀 ↑_m ω) → ((𝑎‘𝐾) ∈ 𝑀 ∧ (𝑎‘𝐿) ∈ 𝑀)))
66	65	imp 405	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘𝐾) ∈ 𝑀 ∧ (𝑎‘𝐿) ∈ 𝑀))
67		brxp 5724	. . . . . . . . . 10 ⊢ ((𝑎‘𝐾)(𝑀 × 𝑀)(𝑎‘𝐿) ↔ ((𝑎‘𝐾) ∈ 𝑀 ∧ (𝑎‘𝐿) ∈ 𝑀))
68	66, 67	sylibr 233	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → (𝑎‘𝐾)(𝑀 × 𝑀)(𝑎‘𝐿))
69	68	biantrud 530	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘𝐾) E (𝑎‘𝐿) ↔ ((𝑎‘𝐾) E (𝑎‘𝐿) ∧ (𝑎‘𝐾)(𝑀 × 𝑀)(𝑎‘𝐿))))
70		fvex 6903	. . . . . . . . 9 ⊢ (𝑎‘𝐿) ∈ V
71	70	epeli 5581	. . . . . . . 8 ⊢ ((𝑎‘𝐾) E (𝑎‘𝐿) ↔ (𝑎‘𝐾) ∈ (𝑎‘𝐿))
72	69, 71	bitr3di 285	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → (((𝑎‘𝐾) E (𝑎‘𝐿) ∧ (𝑎‘𝐾)(𝑀 × 𝑀)(𝑎‘𝐿)) ↔ (𝑎‘𝐾) ∈ (𝑎‘𝐿)))
73	56, 72	bitrid 282	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿) ↔ (𝑎‘𝐾) ∈ (𝑎‘𝐿)))
74	73	notbid 317	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → (¬ (𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿) ↔ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿)))
75	55, 74	orbi12d 915	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((¬ (𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿)) ↔ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))))
76	75	rabbidva 3437	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)( E ∩ (𝑀 × 𝑀))(𝑎‘𝐿))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))})
77	35, 76	eqtrd 2770	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘1_o)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))})
78	6, 33, 77	3eqtrd 2774	1 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat_∈ 𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 {crab 3430 Vcvv 3472 ∩ cin 3946 𝒫 cpw 4601 ∪ ciun 4996 class class class wbr 5147 E cep 5578 × cxp 5673 dom cdm 5675 Fun wfun 6536 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ωcom 7857 1_oc1o 8461 ↑_m cmap 8822 ∈_𝑔cgoe 34622 ⊼_𝑔cgna 34623 Sat csat 34625 Fmlacfmla 34626 Sat_∈ csate 34627

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-ac2 10460

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-ac 10113 df-goel 34629 df-gona 34630 df-goal 34631 df-sat 34632 df-sate 34633 df-fmla 34634

This theorem is referenced by: elnanelprv 34718

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