satefvfmla0 - Mathbox for Mario Carneiro

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Theorem satefvfmla0 33065

Description: The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.)

**Assertion**
Ref	Expression
satefvfmla0	⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋)))})

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.

Step	Hyp	Ref	Expression
1		satefv 33061	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2		incom 4105	. . . . . . . . 9 ⊢ ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3		sqxpexg 7529	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		inex1g 5201	. . . . . . . . . 10 ⊢ ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
6	2, 5	eqeltrid 2838	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
7	6	ancli 552	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
8	7	adantr 484	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9		satom 33003	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
11	10	fveq1d 6708	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12		satfun 33058	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑_m ω))
13	8, 12	syl 17	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑_m ω))
14	13	ffund 6538	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
15	10	eqcomd 2740	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
16	15	funeqd 6391	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) ↔ Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)))
17	14, 16	mpbird 260	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
18		peano1 7656	. . . . . 6 ⊢ ∅ ∈ ω
19	18	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
20	18	a1i 11	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ∅ ∈ ω)
21		satfdmfmla 33047	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
22	6, 20, 21	mpd3an23 1465	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
23	22	eqcomd 2740	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
24	23	eleq2d 2819	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
25	24	biimpa 480	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26		eqid 2734	. . . . . 6 ⊢ ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
27	26	fviunfun 7707	. . . . 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))
28	17, 19, 25, 27	syl3anc 1373	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
29	11, 28	eqtrd 2774	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30		simpl 486	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑀 ∈ 𝑉)
31	6	adantr 484	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32		simpr 488	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33		eqid 2734	. . . . . 6 ⊢ (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
34	33	satfv0fvfmla0 33060	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋)))})
35	30, 31, 32, 34	syl3anc 1373	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋)))})
36		elmapi 8519	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → 𝑎:ω⟶𝑀)
37		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38		fmla0xp 33030	. . . . . . . . . . . . . . . 16 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
39	38	eleq2i 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40		elxp 5563	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ({∅} × (ω × ω)) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
41	39, 40	bitri 278	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
42		xp1st 7782	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (1^st ‘𝑦) ∈ ω)
43	42	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘𝑦) ∈ ω)
44		vex 3405	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
45		vex 3405	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
46	44, 45	op2ndd 7761	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
47	46	fveq2d 6710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (1^st ‘(2^nd ‘𝑋)) = (1^st ‘𝑦))
48	47	eleq1d 2818	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
49	48	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
50	43, 49	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
51	50	exlimivv 1940	. . . . . . . . . . . . . 14 ⊢ (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
52	41, 51	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Fmla‘∅) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
53	52	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
54	37, 53	ffvelrnd 6894	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀)
55		xp2nd 7783	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (2^nd ‘𝑦) ∈ ω)
56	55	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘𝑦) ∈ ω)
57	46	fveq2d 6710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘(2^nd ‘𝑋)) = (2^nd ‘𝑦))
58	57	eleq1d 2818	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
59	58	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
60	56, 59	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
61	60	exlimivv 1940	. . . . . . . . . . . . . 14 ⊢ (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
62	41, 61	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Fmla‘∅) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
63	62	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
64	37, 63	ffvelrnd 6894	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀)
65	54, 64	jca 515	. . . . . . . . . 10 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → ((𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀 ∧ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀))
66	65	ex 416	. . . . . . . . 9 ⊢ (𝑎:ω⟶𝑀 → ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀 ∧ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀)))
67	36, 66	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀 ∧ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀)))
68	67	impcom 411	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀 ∧ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀))
69		brinxp 5616	. . . . . . . 8 ⊢ (((𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀 ∧ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀) → ((𝑎‘(1^st ‘(2^nd ‘𝑋))) E (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋)))))
70	69	bicomd 226	. . . . . . 7 ⊢ (((𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀 ∧ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀) → ((𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘(1^st ‘(2^nd ‘𝑋))) E (𝑎‘(2^nd ‘(2^nd ‘𝑋)))))
71	68, 70	syl 17	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘(1^st ‘(2^nd ‘𝑋))) E (𝑎‘(2^nd ‘(2^nd ‘𝑋)))))
72		fvex 6719	. . . . . . 7 ⊢ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ V
73	72	epeli 5451	. . . . . 6 ⊢ ((𝑎‘(1^st ‘(2^nd ‘𝑋))) E (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋))))
74	71, 73	bitrdi 290	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) ∧ 𝑎 ∈ (𝑀 ↑_m ω)) → ((𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋)))))
75	74	rabbidva 3381	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))( E ∩ (𝑀 × 𝑀))(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋)))})
76	35, 75	eqtrd 2774	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋)))})
77	29, 76	eqtrd 2774	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋)))})
78	1, 77	eqtrd 2774	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ (𝑎‘(2^nd ‘(2^nd ‘𝑋)))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 {crab 3058 Vcvv 3401 ∩ cin 3856 ∅c0 4227 𝒫 cpw 4503 {csn 4531 ⟨cop 4537 ∪ ciun 4894 class class class wbr 5043 E cep 5448 × cxp 5538 dom cdm 5540 Fun wfun 6363 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ωcom 7633 1^st c1st 7748 2^nd c2nd 7749 ↑_m cmap 8497 Sat csat 32983 Fmlacfmla 32984 Sat_∈ csate 32985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-ac2 10060

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-card 9538 df-ac 9713 df-goel 32987 df-gona 32988 df-goal 32989 df-sat 32990 df-sate 32991 df-fmla 32992

This theorem is referenced by: sategoelfvb 33066 prv1n 33078

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