satefvfmla0 - Mathbox for Mario Carneiro

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Theorem satefvfmla0 35405

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 35401	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2		incom 4172	. . . . . . . . 9 ⊢ ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3		sqxpexg 7731	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		inex1g 5274	. . . . . . . . . 10 ⊢ ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
6	2, 5	eqeltrid 2832	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
7	6	ancli 548	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
8	7	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9		satom 35343	. . . . . 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 6860	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12		satfun 35398	. . . . . . . 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 6692	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
15	10	eqcomd 2735	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
16	15	funeqd 6538	. . . . . 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 257	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
18		peano1 7865	. . . . . 6 ⊢ ∅ ∈ ω
19	18	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
20	18	a1i 11	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ∅ ∈ ω)
21		satfdmfmla 35387	. . . . . . . . 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 2735	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
24	23	eleq2d 2814	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
25	24	biimpa 476	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26		eqid 2729	. . . . . 6 ⊢ ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
27	26	fviunfun 7923	. . . . 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 2764	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30		simpl 482	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑀 ∈ 𝑉)
31	6	adantr 480	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32		simpr 484	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33		eqid 2729	. . . . . 6 ⊢ (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
34	33	satfv0fvfmla0 35400	. . . . 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 8822	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → 𝑎:ω⟶𝑀)
37		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38		fmla0xp 35370	. . . . . . . . . . . . . . . 16 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
39	38	eleq2i 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40		elxp 5661	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ({∅} × (ω × ω)) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
41	39, 40	bitri 275	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
42		xp1st 8000	. . . . . . . . . . . . . . . . 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 3451	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
45		vex 3451	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
46	44, 45	op2ndd 7979	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
47	46	fveq2d 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (1^st ‘(2^nd ‘𝑋)) = (1^st ‘𝑦))
48	47	eleq1d 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
49	48	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
50	43, 49	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
51	50	exlimivv 1932	. . . . . . . . . . . . . 14 ⊢ (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
52	41, 51	sylbi 217	. . . . . . . . . . . . 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	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀)
55		xp2nd 8001	. . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘(2^nd ‘𝑋)) = (2^nd ‘𝑦))
58	57	eleq1d 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
60	56, 59	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
61	60	exlimivv 1932	. . . . . . . . . . . . . 14 ⊢ (∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
62	41, 61	sylbi 217	. . . . . . . . . . . . 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	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀)
65	54, 64	jca 511	. . . . . . . . . 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 412	. . . . . . . . 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 407	. . . . . . 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 5717	. . . . . . . 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 223	. . . . . . 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 6871	. . . . . . 7 ⊢ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ V
73	72	epeli 5540	. . . . . 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 287	. . . . 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 3412	. . . 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 2764	. . 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 2764	. 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 2764	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 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 {crab 3405 Vcvv 3447 ∩ cin 3913 ∅c0 4296 𝒫 cpw 4563 {csn 4589 ⟨cop 4595 ∪ ciun 4955 class class class wbr 5107 E cep 5537 × cxp 5636 dom cdm 5638 Fun wfun 6505 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ωcom 7842 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8799 Sat csat 35323 Fmlacfmla 35324 Sat_∈ csate 35325

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-ac2 10416

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-card 9892 df-ac 10069 df-goel 35327 df-gona 35328 df-goal 35329 df-sat 35330 df-sate 35331 df-fmla 35332

This theorem is referenced by: sategoelfvb 35406 prv1n 35418

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