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Theorem setsstruct2 17240

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 17215	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2		elin 3920	. . . . . . . . 9 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3		elxp6 8018	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)))
4		eleq1 2850	. . . . . . . . . . . . 13 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ))
5	4	adantr 485	. . . . . . . . . . . 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 1155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
7		simp1l 1215	. . . . . . . . . . . . . . . . . . 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 4533	. . . . . . . . . . . . . . . . . 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 12254	. . . . . . . . . . . . . . . . 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 12254	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℝ)
11		simp1r 1216	. . . . . . . . . . . . . . . . . . 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 4533	. . . . . . . . . . . . . . . . . 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 12254	. . . . . . . . . . . . . . . . 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 12246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝑋) ∈ ℕ → (1^st ‘𝑋) ∈ ℝ)
15	14	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (1^st ‘𝑋) ∈ ℝ)
16		nnre 12246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℝ)
17	15, 16	anim12i 624	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((1^st ‘𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
18	17	3adant2 1148	. . . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . . 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 13221	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ ℝ ∧ (1^st ‘𝑋) ∈ ℝ) → if(𝐼 ≤ (1^st ‘𝑋), 𝐼, (1^st ‘𝑋)) ≤ 𝐼)
21	19, 20	syl 18	. . . . . . . . . . . . . . . . 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 12246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘𝑋) ∈ ℕ → (2^nd ‘𝑋) ∈ ℝ)
23	22	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (2^nd ‘𝑋) ∈ ℝ)
24	23, 16	anim12i 624	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((2^nd ‘𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
25	24	3adant2 1148	. . . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . . 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 13217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ ℝ ∧ (2^nd ‘𝑋) ∈ ℝ) → 𝐼 ≤ if(𝐼 ≤ (2^nd ‘𝑋), (2^nd ‘𝑋), 𝐼))
28	26, 27	syl 18	. . . . . . . . . . . . . . . . 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 11373	. . . . . . . . . . . . . . . 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 5109	. . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . 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 5699	. . . . . . . . . . . . . . 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 4152	. . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 243	. . . . . . . . . . 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 220	. . . . . . . . . 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 412	. . . . . . . . 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 220	. . . . . . . 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 1150	. . . . . . 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 220	. . . . . 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 411	. . . . 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 1148	. . . 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 17216	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V)
45		structn0fun 17217	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
46	44, 45	jca 520	. . . . . 6 ⊢ (𝐺 Struct 𝑋 → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
47	46	3ad2ant1 1150	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
48		simp3 1155	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
49		simp2 1154	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐸 ∈ 𝑉)
50		setsfun0 17238	. . . . 5 ⊢ (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
51	47, 48, 49, 50	syl12anc 849	. . . 4 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52	44	3ad2ant1 1150	. . . . . 6 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐺 ∈ V)
53		setsdm 17236	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝐸 ∈ 𝑉) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
54	52, 49, 53	syl2anc 595	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
55		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (...‘𝑋) = (...‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
56		df-ov 7415	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑋)...(2^nd ‘𝑋)) = (...‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
57	55, 56	eqtr4di 2815	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (...‘𝑋) = ((1^st ‘𝑋)...(2^nd ‘𝑋)))
58	57	sseq2d 3968	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋))))
59	58	adantr 485	. . . . . . . . . . . . . 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 1104	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) ↔ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ))
61		nnz 12618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1^st ‘𝑋) ∈ ℕ → (1^st ‘𝑋) ∈ ℤ)
62		nnz 12618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑋) ∈ ℕ → (2^nd ‘𝑋) ∈ ℤ)
63		nnz 12618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
64	61, 62, 63	3anim123i 1168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1^st ‘𝑋) ∈ ℤ ∧ (2^nd ‘𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
65		ssfzunsnext 13604	. . . . . . . . . . . . . . . . . . . . 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 7415	. . . . . . . . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . 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	biimtrrid 246	. . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 243	. . . . . . . . . . . . 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 220	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 220	. . . . . . . . . 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 411	. . . . . . . . 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 1148	. . . . . . . 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 220	. . . . . . 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 411	. . . . . 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 1148	. . . . 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 3970	. . . 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 17215	. . . 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 1361	. . 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 485	. 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 5112	. . 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 486	. 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 260	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 Vcvv 3454 ∖ cdif 3901 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 ifcif 4486 {csn 4588 ⟨cop 4594 class class class wbr 5108 × cxp 5658 dom cdm 5660 Fun wfun 6530 ‘cfv 6536 (class class class)co 7412 1^st c1st 7982 2^nd c2nd 7983 ℝcr 11105 ≤ cle 11250 ℕcn 12239 ℤcz 12597 ...cfz 13541 Struct cstr 17212 sSet csts 17229

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 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-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-n0 12511 df-z 12598 df-uz 12869 df-fz 13542 df-struct 17213 df-sets 17230

This theorem is used by: setsexstruct2 17241 setsstruct 17242

W3C validator