satefvfmla0 - Mathbox for Mario Carneiro

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Theorem satefvfmla0 35631

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 35627	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat_∈ 𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋))
2		incom 4163	. . . . . . . . 9 ⊢ ( E ∩ (𝑀 × 𝑀)) = ((𝑀 × 𝑀) ∩ E )
3		sqxpexg 7710	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		inex1g 5266	. . . . . . . . . 10 ⊢ ((𝑀 × 𝑀) ∈ V → ((𝑀 × 𝑀) ∩ E ) ∈ V)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ((𝑀 × 𝑀) ∩ E ) ∈ V)
6	2, 5	eqeltrid 2841	. . . . . . . 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 35569	. . . . . 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 6844	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑋) = (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋))
12		satfun 35624	. . . . . . . 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 6674	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
15	10	eqcomd 2743	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
16	15	funeqd 6522	. . . . . 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 7841	. . . . . 6 ⊢ ∅ ∈ ω
19	18	a1i 11	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → ∅ ∈ ω)
20	18	a1i 11	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → ∅ ∈ ω)
21		satfdmfmla 35613	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ ( E ∩ (𝑀 × 𝑀)) ∈ V ∧ ∅ ∈ ω) → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
22	6, 20, 21	mpd3an23 1466	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅) = (Fmla‘∅))
23	22	eqcomd 2743	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (Fmla‘∅) = dom ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅))
24	23	eleq2d 2823	. . . . . 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 2737	. . . . . 6 ⊢ ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖) = ∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)
27	26	fviunfun 7899	. . . . 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 1374	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (∪ 𝑖 ∈ ω ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘𝑖)‘𝑋) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘∅)‘𝑋))
29	11, 28	eqtrd 2772	. . 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 2737	. . . . . 6 ⊢ (𝑀 Sat ( E ∩ (𝑀 × 𝑀))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))
34	33	satfv0fvfmla0 35626	. . . . 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 1374	. . . 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 8798	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑀 ↑_m ω) → 𝑎:ω⟶𝑀)
37		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → 𝑎:ω⟶𝑀)
38		fmla0xp 35596	. . . . . . . . . . . . . . . 16 ⊢ (Fmla‘∅) = ({∅} × (ω × ω))
39	38	eleq2i 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ ({∅} × (ω × ω)))
40		elxp 5655	. . . . . . . . . . . . . . 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 7975	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (1^st ‘𝑦) ∈ ω)
43	42	ad2antll 730	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (1^st ‘𝑦) ∈ ω)
44		vex 3446	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
45		vex 3446	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
46	44, 45	op2ndd 7954	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
47	46	fveq2d 6846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (1^st ‘(2^nd ‘𝑋)) = (1^st ‘𝑦))
48	47	eleq1d 2822	. . . . . . . . . . . . . . . . 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 1934	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (1^st ‘(2^nd ‘𝑋)) ∈ ω)
54	37, 53	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (𝑎‘(1^st ‘(2^nd ‘𝑋))) ∈ 𝑀)
55		xp2nd 7976	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ω × ω) → (2^nd ‘𝑦) ∈ ω)
56	55	ad2antll 730	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {∅} ∧ 𝑦 ∈ (ω × ω))) → (2^nd ‘𝑦) ∈ ω)
57	46	fveq2d 6846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = ⟨𝑥, 𝑦⟩ → (2^nd ‘(2^nd ‘𝑋)) = (2^nd ‘𝑦))
58	57	eleq1d 2822	. . . . . . . . . . . . . . . . 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 1934	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ ((𝑎:ω⟶𝑀 ∧ (𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅))) → (2^nd ‘(2^nd ‘𝑋)) ∈ ω)
64	37, 63	ffvelcdmd 7039	. . . . . . . . . . 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 5711	. . . . . . . 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 6855	. . . . . . 7 ⊢ (𝑎‘(2^nd ‘(2^nd ‘𝑋))) ∈ V
73	72	epeli 5534	. . . . . 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 3407	. . . 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 2772	. . 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 2772	. 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 2772	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 1542 ∃wex 1781 ∈ wcel 2114 {crab 3401 Vcvv 3442 ∩ cin 3902 ∅c0 4287 𝒫 cpw 4556 {csn 4582 ⟨cop 4588 ∪ ciun 4948 class class class wbr 5100 E cep 5531 × cxp 5630 dom cdm 5632 Fun wfun 6494 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ωcom 7818 1^st c1st 7941 2^nd c2nd 7942 ↑_m cmap 8775 Sat csat 35549 Fmlacfmla 35550 Sat_∈ csate 35551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-ac2 10385

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-ac 10038 df-goel 35553 df-gona 35554 df-goal 35555 df-sat 35556 df-sate 35557 df-fmla 35558

This theorem is referenced by: sategoelfvb 35632 prv1n 35644

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