satefvfmla0 - Mathbox for Mario Carneiro

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Theorem satefvfmla0 32689

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 32685	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2		incom 4175	. . . . . . . . 9 ⊢ ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3		sqxpexg 7474	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		inex1g 5220	. . . . . . . . . 10 ⊢ ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
6	2, 5	eqeltrid 2916	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → ( E ∩ (𝑀 × 𝑀)) ∈ V)
7	6	ancli 551	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
8	7	adantr 483	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V))
9		satom 32627	. . . . . 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 6669	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12		satfun 32682	. . . . . . . 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 6515	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
15	10	eqcomd 2826	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
16	15	funeqd 6374	. . . . . 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 259	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖))
18		peano1 7598	. . . . . 6 ⊢ ∅ ∈ ω
19	18	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
20	18	a1i 11	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ∅ ∈ ω)
21		satfdmfmla 32671	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
22	6, 20, 21	mpd3an23 1458	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
23	22	eqcomd 2826	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
24	23	eleq2d 2897	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)))
25	24	biimpa 479	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
26		eqid 2820	. . . . . 6 ⊢ ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
27	26	fviunfun 7643	. . . . 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 1366	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
29	11, 28	eqtrd 2855	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
30		simpl 485	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑀 ∈ 𝑉)
31	6	adantr 483	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ( E ∩ (𝑀 × 𝑀)) ∈ V)
32		simpr 487	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
33		eqid 2820	. . . . . 6 ⊢ (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
34	33	satfv0fvfmla0 32684	. . . . 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 1366	. . . 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 8425	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → 𝑎:ω⟶𝑀)
37		simpl 485	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38		fmla0xp 32654	. . . . . . . . . . . . . . . 16 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
39	38	eleq2i 2903	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40		elxp 5575	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ({∅} × (ω × ω)) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
41	39, 40	bitri 277	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ ∃𝑥∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))))
42		xp1st 7718	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (1^st ‘𝑦) ∈ ω)
43	42	ad2antll 727	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘𝑦) ∈ ω)
44		vex 3496	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
45		vex 3496	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
46	44, 45	op2ndd 7697	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
47	46	fveq2d 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (1^st ‘(2^nd ‘𝑋)) = (1^st ‘𝑦))
48	47	eleq1d 2896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
49	48	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((1^st ‘(2^nd ‘𝑋)) ∈ ω ↔ (1^st ‘𝑦) ∈ ω))
50	43, 49	mpbird 259	. . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Fmla‘∅) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
53	52	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
54	37, 53	ffvelrnd 6849	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀)
55		xp2nd 7719	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (2^nd ‘𝑦) ∈ ω)
56	55	ad2antll 727	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘𝑦) ∈ ω)
57	46	fveq2d 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘(2^nd ‘𝑋)) = (2^nd ‘𝑦))
58	57	eleq1d 2896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
59	58	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → ((2^nd ‘(2^nd ‘𝑋)) ∈ ω ↔ (2^nd ‘𝑦) ∈ ω))
60	56, 59	mpbird 259	. . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ (Fmla‘∅) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
63	62	ad2antll 727	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
64	37, 63	ffvelrnd 6849	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ 𝑀)
65	54, 64	jca 514	. . . . . . . . . 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 415	. . . . . . . . 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 410	. . . . . . 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 5627	. . . . . . . 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 225	. . . . . . 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 6680	. . . . . . 7 ⊢ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ V
73	72	epeli 5465	. . . . . 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	syl6bb 289	. . . . 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 3477	. . . 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 2855	. . 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 2855	. 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 2855	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 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 {crab 3141 Vcvv 3493 ∩ cin 3932 ∅c0 4288 𝒫 cpw 4536 {csn 4564 ⟨cop 4570 ∪ ciun 4916 class class class wbr 5063 E cep 5461 × cxp 5550 dom cdm 5552 Fun wfun 6346 ⟶wf 6348 ‘cfv 6352 (class class class)co 7153 ωcom 7577 1^st c1st 7684 2^nd c2nd 7685 ↑_m cmap 8403 Sat csat 32607 Fmlacfmla 32608 Sat_∈ csate 32609

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-inf2 9101 ax-ac2 9882

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-er 8286 df-map 8405 df-en 8507 df-dom 8508 df-sdom 8509 df-card 9365 df-ac 9539 df-goel 32611 df-gona 32612 df-goal 32613 df-sat 32614 df-sate 32615 df-fmla 32616

This theorem is referenced by: sategoelfvb 32690 prv1n 32702

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