satefvfmla0 - Mathbox for Mario Carneiro

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Theorem satefvfmla0 35999

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 35995	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2		incom 4158	. . . . . . . . 9 ⊢ ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3		sqxpexg 7757	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		inex1g 5286	. . . . . . . . . 10 ⊢ ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
5	3, 4	syl 18	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
6	2, 5	eqeltrid 2866	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
7	6	ancli 558	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
8	7	adantr 486	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9		satom 35937	. . . . . 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 18	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
11	10	fveq1d 6884	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12		satfun 35992	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑_m ω))
13	8, 12	syl 18	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑_m ω))
14	13	ffund 6711	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
15	10	eqcomd 2768	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
16	15	funeqd 6559	. . . . . 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 7888	. . . . . 6 ⊢ ∅ ∈ ω
19	18	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
20	18	a1i 11	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ∅ ∈ ω)
21		satfdmfmla 35981	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
22	6, 20, 21	mpd3an23 1492	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
23	22	eqcomd 2768	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
24	23	eleq2d 2848	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
25	24	biimpa 482	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26		eqid 2762	. . . . . 6 ⊢ ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
27	26	fviunfun 7945	. . . . 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 1398	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
29	11, 28	eqtrd 2797	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30		simpl 488	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑀 ∈ 𝑉)
31	6	adantr 486	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32		simpr 490	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33		eqid 2762	. . . . . 6 ⊢ (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
34	33	satfv0fvfmla0 35994	. . . . 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 1398	. . . 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 8851	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → 𝑎:ω⟶𝑀)
37		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38		fmla0xp 35964	. . . . . . . . . . . . . . . 16 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
39	38	eleq2i 2854	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40		elxp 5682	. . . . . . . . . . . . . . 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 8021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (1^st ‘𝑦) ∈ ω)
43	42	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘𝑦) ∈ ω)
44		vex 3457	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
45		vex 3457	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
46	44, 45	op2ndd 8000	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
47	46	fveq2d 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (1^st ‘(2^nd ‘𝑋)) = (1^st ‘𝑦))
48	47	eleq1d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
49	48	adantr 486	. . . . . . . . . . . . . . . 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 1965	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
54	37, 53	ffvelcdmd 7081	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀)
55		xp2nd 8022	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (2^nd ‘𝑦) ∈ ω)
56	55	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘𝑦) ∈ ω)
57	46	fveq2d 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘(2^nd ‘𝑋)) = (2^nd ‘𝑦))
58	57	eleq1d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
59	58	adantr 486	. . . . . . . . . . . . . . . 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 1965	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
64	37, 63	ffvelcdmd 7081	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀)
65	54, 64	jca 521	. . . . . . . . . 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 418	. . . . . . . . 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 18	. . . . . . . 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 413	. . . . . . 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 5738	. . . . . . . 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 18	. . . . . 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 6895	. . . . . . 7 ⊢ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ V
73	72	epeli 5561	. . . . . 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 3420	. . . 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 2797	. . 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 2797	. 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 2797	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 {crab 3414 Vcvv 3453 ∩ cin 3901 ∅c0 4282 𝒫 cpw 4560 {csn 4587 ⟨cop 4593 ∪ ciun 4954 class class class wbr 5107 E cep 5558 × cxp 5657 dom cdm 5659 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ωcom 7865 1^st c1st 7987 2^nd c2nd 7988 ↑_m cmap 8829 Sat csat 35917 Fmlacfmla 35918 Sat_∈ csate 35919

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-ac2 10468

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9947 df-ac 10122 df-goel 35921 df-gona 35922 df-goal 35923 df-sat 35924 df-sate 35925 df-fmla 35926

This theorem is used by: sategoelfvb 36000 prv1n 36012

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