setsstruct2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > setsstruct2		Structured version Visualization version GIF version

Theorem setsstruct2 16875

Description: An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)

**Assertion**
Ref	Expression
setsstruct2	⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)

**Proof of Theorem setsstruct2**
Step	Hyp	Ref	Expression
1		isstruct2 16850	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2		elin 3903	. . . . . . . . 9 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3		elxp6 7865	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)))
4		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ))
5	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)) → (𝑋 ∈ ≤ ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ))
6		simp3 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
7		simp1l 1196	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (1^st ‘𝑋) ∈ ℕ)
8	6, 7	ifcld 4505	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ∈ ℕ)
9	8	nnred 11988	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ∈ ℝ)
10	6	nnred 11988	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℝ)
11		simp1r 1197	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (2^nd ‘𝑋) ∈ ℕ)
12	11, 6	ifcld 4505	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼) ∈ ℕ)
13	12	nnred 11988	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼) ∈ ℝ)
14		nnre 11980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝑋) ∈ ℕ → (1^st ‘𝑋) ∈ ℝ)
15	14	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (1^st ‘𝑋) ∈ ℝ)
16		nnre 11980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℝ)
17	15, 16	anim12i 613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((1^st ‘𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
18	17	3adant2 1130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((1^st ‘𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
19	18	ancomd 462	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (1^st ‘𝑋) ∈ ℝ))
20		min1 12923	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ ℝ ∧ (1^st ‘𝑋) ∈ ℝ) → if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ≤ 𝐼)
21	19, 20	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ≤ 𝐼)
22		nnre 11980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘𝑋) ∈ ℕ → (2^nd ‘𝑋) ∈ ℝ)
23	22	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (2^nd ‘𝑋) ∈ ℝ)
24	23, 16	anim12i 613	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((2^nd ‘𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
25	24	3adant2 1130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((2^nd ‘𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
26	25	ancomd 462	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (2^nd ‘𝑋) ∈ ℝ))
27		max1 12919	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ ℝ ∧ (2^nd ‘𝑋) ∈ ℝ) → 𝐼 ≤ if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼))
28	26, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ≤ if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼))
29	9, 10, 13, 21, 28	letrd 11132	. . . . . . . . . . . . . . . 16 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ≤ if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼))
30		df-br 5075	. . . . . . . . . . . . . . . 16 ⊢ (if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ≤ if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼) ↔ ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ≤ )
31	29, 30	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ≤ )
32	8, 12	opelxpd 5627	. . . . . . . . . . . . . . 15 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ (ℕ × ℕ))
33	31, 32	elind 4128	. . . . . . . . . . . . . 14 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
34	33	3exp 1118	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
35	34	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)) → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
36	5, 35	sylbid 239	. . . . . . . . . . 11 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)) → (𝑋 ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
37	3, 36	sylbi 216	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℕ × ℕ) → (𝑋 ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
38	37	impcom 408	. . . . . . . . 9 ⊢ ((𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
39	2, 38	sylbi 216	. . . . . . . 8 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
40	39	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
41	1, 40	sylbi 216	. . . . . 6 ⊢ (𝐺 Struct 𝑋 → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
42	41	imp 407	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
43	42	3adant2 1130	. . . 4 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
44		structex 16851	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V)
45		structn0fun 16852	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
46	44, 45	jca 512	. . . . . 6 ⊢ (𝐺 Struct 𝑋 → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
47	46	3ad2ant1 1132	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
48		simp3 1137	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
49		simp2 1136	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐸 ∈ 𝑉)
50		setsfun0 16873	. . . . 5 ⊢ (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
51	47, 48, 49, 50	syl12anc 834	. . . 4 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52	44	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐺 ∈ V)
53		setsdm 16871	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐸 ∈ 𝑉) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
54	52, 49, 53	syl2anc 584	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
55		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (...‘𝑋) = (...‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
56		df-ov 7278	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑋)...(2^nd ‘𝑋)) = (...‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
57	55, 56	eqtr4di 2796	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (...‘𝑋) = ((1^st ‘𝑋)...(2^nd ‘𝑋)))
58	57	sseq2d 3953	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋))))
59	58	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋))))
60		df-3an 1088	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) ↔ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ))
61		nnz 12342	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1^st ‘𝑋) ∈ ℕ → (1^st ‘𝑋) ∈ ℤ)
62		nnz 12342	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑋) ∈ ℕ → (2^nd ‘𝑋) ∈ ℤ)
63		nnz 12342	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
64	61, 62, 63	3anim123i 1150	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1^st ‘𝑋) ∈ ℤ ∧ (2^nd ‘𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
65		ssfzunsnext 13301	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) ∧ ((1^st ‘𝑋) ∈ ℤ ∧ (2^nd ‘𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋))...if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)))
66		df-ov 7278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋))...if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)) = (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)
67	65, 66	sseqtrdi 3971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) ∧ ((1^st ‘𝑋) ∈ ℤ ∧ (2^nd ‘𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
68	64, 67	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
69	68	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) → (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)))
70	60, 69	syl5bir 242	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) → ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)))
71	70	expd 416	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) → (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
72	71	com12 32	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
73	72	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
74	59, 73	sylbid 239	. . . . . . . . . . . . 13 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
75	3, 74	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (ℕ × ℕ) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
76	75	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
77	2, 76	sylbi 216	. . . . . . . . . 10 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))))
78	77	imp 407	. . . . . . . . 9 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)))
79	78	3adant2 1130	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)))
80	1, 79	sylbi 216	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)))
81	80	imp 407	. . . . . 6 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
82	81	3adant2 1130	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
83	54, 82	eqsstrd 3959	. . . 4 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
84		isstruct2 16850	. . . 4 ⊢ ((𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ↔ (⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)))
85	43, 51, 83, 84	syl3anbrc 1342	. . 3 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)
86	85	adantr 481	. 2 ⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩)
87		breq2 5078	. . 3 ⊢ (𝑌 = ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩ → ((𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌 ↔ (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
88	87	adantl 482	. 2 ⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩) → ((𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌 ↔ (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩))
89	86, 88	mpbird 256	1 ⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)), if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 ifcif 4459 {csn 4561 ⟨cop 4567 class class class wbr 5074 × cxp 5587 dom cdm 5589 Fun wfun 6427 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 ℝcr 10870 ≤ cle 11010 ℕcn 11973 ℤcz 12319 ...cfz 13239 Struct cstr 16847 sSet csts 16864

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865

This theorem is referenced by: setsexstruct2 16876 setsstruct 16877

W3C validator