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Theorem setsstruct2 17144

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 17119	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2		elin 3930	. . . . . . . . 9 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3		elxp6 8002	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ)))
4		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ))
5	4	adantr 480	. . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
7		simp1l 1198	. . . . . . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . . . . . . 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 12201	. . . . . . . . . . . . . . . . 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 12201	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) ∧ ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℝ)
11		simp1r 1199	. . . . . . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . . . . . . 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 12201	. . . . . . . . . . . . . . . . 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 12193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝑋) ∈ ℕ → (1^st ‘𝑋) ∈ ℝ)
15	14	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ) → (1^st ‘𝑋) ∈ ℝ)
16		nnre 12193	. . . . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . 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 13149	. . . . . . . . . . . . . . . . . 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 12193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2^nd ‘𝑋) ∈ ℕ → (2^nd ‘𝑋) ∈ ℝ)
23	22	adantl 481	. . . . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . 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 13145	. . . . . . . . . . . . . . . . . 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 11331	. . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . 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 5677	. . . . . . . . . . . . . . 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 4163	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 240	. . . . . . . . . . 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 217	. . . . . . . . . 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 407	. . . . . . . . 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 217	. . . . . . . 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 1133	. . . . . . 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 217	. . . . . 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 406	. . . . 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 1131	. . . 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 17120	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V)
45		structn0fun 17121	. . . . . . 7 ⊢ (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
46	44, 45	jca 511	. . . . . 6 ⊢ (𝐺 Struct 𝑋 → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
47	46	3ad2ant1 1133	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
48		simp3 1138	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
49		simp2 1137	. . . . 5 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐸 ∈ 𝑉)
50		setsfun0 17142	. . . . 5 ⊢ (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸 ∈ 𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
51	47, 48, 49, 50	syl12anc 836	. . . 4 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52	44	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → 𝐺 ∈ V)
53		setsdm 17140	. . . . . 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 6858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (...‘𝑋) = (...‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩))
56		df-ov 7390	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑋)...(2^nd ‘𝑋)) = (...‘⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
57	55, 56	eqtr4di 2782	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (...‘𝑋) = ((1^st ‘𝑋)...(2^nd ‘𝑋)))
58	57	sseq2d 3979	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1^st ‘𝑋)...(2^nd ‘𝑋))))
59	58	adantr 480	. . . . . . . . . . . . . 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 12550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1^st ‘𝑋) ∈ ℕ → (1^st ‘𝑋) ∈ ℤ)
62		nnz 12550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2^nd ‘𝑋) ∈ ℕ → (2^nd ‘𝑋) ∈ ℤ)
63		nnz 12550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
64	61, 62, 63	3anim123i 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1^st ‘𝑋) ∈ ℕ ∧ (2^nd ‘𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1^st ‘𝑋) ∈ ℤ ∧ (2^nd ‘𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
65		ssfzunsnext 13530	. . . . . . . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 240	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 217	. . . . . . . . . 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 406	. . . . . . . . 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 1131	. . . . . . . 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 217	. . . . . . 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 406	. . . . . 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 1131	. . . . 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 3981	. . . 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 17119	. . . 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 1344	. . 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 480	. 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 5111	. . 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 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 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 257	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 ifcif 4488 {csn 4589 ⟨cop 4595 class class class wbr 5107 × cxp 5636 dom cdm 5638 Fun wfun 6505 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 ℝcr 11067 ≤ cle 11209 ℕcn 12186 ℤcz 12529 ...cfz 13468 Struct cstr 17116 sSet csts 17133

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-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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-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-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-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-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-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134

This theorem is referenced by: setsexstruct2 17145 setsstruct 17146

W3C validator