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Theorem setsstruct2 17108
Description: An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)
Assertion
Ref Expression
setsstruct2 (((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)

Proof of Theorem setsstruct2
StepHypRef Expression
1 isstruct2 17083 . . . . . . 7 (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2 elin 3957 . . . . . . . . 9 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3 elxp6 8003 . . . . . . . . . . 11 (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)))
4 eleq1 2813 . . . . . . . . . . . . 13 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
54adantr 480 . . . . . . . . . . . 12 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
6 simp3 1135 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
7 simp1l 1194 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (1st𝑋) ∈ ℕ)
86, 7ifcld 4567 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℕ)
98nnred 12225 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℝ)
106nnred 12225 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℝ)
11 simp1r 1195 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (2nd𝑋) ∈ ℕ)
1211, 6ifcld 4567 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℕ)
1312nnred 12225 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℝ)
14 nnre 12217 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℝ)
1514adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℝ)
16 nnre 12217 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℝ)
1715, 16anim12i 612 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
18173adant2 1128 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
1918ancomd 461 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (1st𝑋) ∈ ℝ))
20 min1 13166 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∈ ℝ ∧ (1st𝑋) ∈ ℝ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ 𝐼)
2119, 20syl 17 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ 𝐼)
22 nnre 12217 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℝ)
2322adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ∈ ℝ)
2423, 16anim12i 612 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((2nd𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
25243adant2 1128 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((2nd𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
2625ancomd 461 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (2nd𝑋) ∈ ℝ))
27 max1 13162 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∈ ℝ ∧ (2nd𝑋) ∈ ℝ) → 𝐼 ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼))
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼))
299, 10, 13, 21, 28letrd 11369 . . . . . . . . . . . . . . . 16 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼))
30 df-br 5140 . . . . . . . . . . . . . . . 16 (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ↔ ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ≤ )
3129, 30sylib 217 . . . . . . . . . . . . . . 15 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ≤ )
328, 12opelxpd 5706 . . . . . . . . . . . . . . 15 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ (ℕ × ℕ))
3331, 32elind 4187 . . . . . . . . . . . . . 14 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
34333exp 1116 . . . . . . . . . . . . 13 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
3534adantl 481 . . . . . . . . . . . 12 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
365, 35sylbid 239 . . . . . . . . . . 11 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (𝑋 ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
373, 36sylbi 216 . . . . . . . . . 10 (𝑋 ∈ (ℕ × ℕ) → (𝑋 ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
3837impcom 407 . . . . . . . . 9 ((𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
392, 38sylbi 216 . . . . . . . 8 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
40393ad2ant1 1130 . . . . . . 7 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
411, 40sylbi 216 . . . . . 6 (𝐺 Struct 𝑋 → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
4241imp 406 . . . . 5 ((𝐺 Struct 𝑋𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
43423adant2 1128 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
44 structex 17084 . . . . . . 7 (𝐺 Struct 𝑋𝐺 ∈ V)
45 structn0fun 17085 . . . . . . 7 (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
4644, 45jca 511 . . . . . 6 (𝐺 Struct 𝑋 → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
47463ad2ant1 1130 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
48 simp3 1135 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
49 simp2 1134 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐸𝑉)
50 setsfun0 17106 . . . . 5 (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
5147, 48, 49, 50syl12anc 834 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52443ad2ant1 1130 . . . . . 6 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐺 ∈ V)
53 setsdm 17104 . . . . . 6 ((𝐺 ∈ V ∧ 𝐸𝑉) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
5452, 49, 53syl2anc 583 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
55 fveq2 6882 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = (...‘⟨(1st𝑋), (2nd𝑋)⟩))
56 df-ov 7405 . . . . . . . . . . . . . . . . 17 ((1st𝑋)...(2nd𝑋)) = (...‘⟨(1st𝑋), (2nd𝑋)⟩)
5755, 56eqtr4di 2782 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = ((1st𝑋)...(2nd𝑋)))
5857sseq2d 4007 . . . . . . . . . . . . . . 15 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋))))
5958adantr 480 . . . . . . . . . . . . . 14 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋))))
60 df-3an 1086 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) ↔ (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ))
61 nnz 12577 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℤ)
62 nnz 12577 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
63 nnz 12577 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
6461, 62, 633anim123i 1148 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
65 ssfzunsnext 13544 . . . . . . . . . . . . . . . . . . . . 21 ((dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) ∧ ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋))...if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)))
66 df-ov 7405 . . . . . . . . . . . . . . . . . . . . 21 (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋))...if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)) = (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
6765, 66sseqtrdi 4025 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) ∧ ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
6864, 67sylan2 592 . . . . . . . . . . . . . . . . . . 19 ((dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
6968ex 412 . . . . . . . . . . . . . . . . . 18 (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
7060, 69biimtrrid 242 . . . . . . . . . . . . . . . . 17 (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
7170expd 415 . . . . . . . . . . . . . . . 16 (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7271com12 32 . . . . . . . . . . . . . . 15 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7372adantl 481 . . . . . . . . . . . . . 14 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7459, 73sylbid 239 . . . . . . . . . . . . 13 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
753, 74sylbi 216 . . . . . . . . . . . 12 (𝑋 ∈ (ℕ × ℕ) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7675adantl 481 . . . . . . . . . . 11 ((𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
772, 76sylbi 216 . . . . . . . . . 10 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7877imp 406 . . . . . . . . 9 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
79783adant2 1128 . . . . . . . 8 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
801, 79sylbi 216 . . . . . . 7 (𝐺 Struct 𝑋 → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
8180imp 406 . . . . . 6 ((𝐺 Struct 𝑋𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
82813adant2 1128 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
8354, 82eqsstrd 4013 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
84 isstruct2 17083 . . . 4 ((𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ↔ (⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}) ∧ dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
8543, 51, 83, 84syl3anbrc 1340 . . 3 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
8685adantr 480 . 2 (((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
87 breq2 5143 . . 3 (𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ → ((𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌 ↔ (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
8887adantl 481 . 2 (((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → ((𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌 ↔ (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
8986, 88mpbird 257 1 (((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  cdif 3938  cun 3939  cin 3940  wss 3941  c0 4315  ifcif 4521  {csn 4621  cop 4627   class class class wbr 5139   × cxp 5665  dom cdm 5667  Fun wfun 6528  cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  cr 11106  cle 11247  cn 12210  cz 12556  ...cfz 13482   Struct cstr 17080   sSet csts 17097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-nn 12211  df-n0 12471  df-z 12557  df-uz 12821  df-fz 13483  df-struct 17081  df-sets 17098
This theorem is referenced by:  setsexstruct2  17109  setsstruct  17110
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