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Theorem setsstruct2 17111
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 17086 . . . . . . 7 (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2 elin 3963 . . . . . . . . 9 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3 elxp6 8011 . . . . . . . . . . 11 (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)))
4 eleq1 2819 . . . . . . . . . . . . 13 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
54adantr 479 . . . . . . . . . . . 12 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
6 simp3 1136 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
7 simp1l 1195 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (1st𝑋) ∈ ℕ)
86, 7ifcld 4573 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℕ)
98nnred 12231 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℝ)
106nnred 12231 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℝ)
11 simp1r 1196 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (2nd𝑋) ∈ ℕ)
1211, 6ifcld 4573 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℕ)
1312nnred 12231 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℝ)
14 nnre 12223 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℝ)
1514adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℝ)
16 nnre 12223 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℝ)
1715, 16anim12i 611 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
18173adant2 1129 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
1918ancomd 460 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (1st𝑋) ∈ ℝ))
20 min1 13172 . . . . . . . . . . . . . . . . . 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 12223 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℝ)
2322adantl 480 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ∈ ℝ)
2423, 16anim12i 611 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((2nd𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
25243adant2 1129 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((2nd𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
2625ancomd 460 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (2nd𝑋) ∈ ℝ))
27 max1 13168 . . . . . . . . . . . . . . . . . 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 11375 . . . . . . . . . . . . . . . 16 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼))
30 df-br 5148 . . . . . . . . . . . . . . . 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 5714 . . . . . . . . . . . . . . 15 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ (ℕ × ℕ))
3331, 32elind 4193 . . . . . . . . . . . . . 14 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
34333exp 1117 . . . . . . . . . . . . 13 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
3534adantl 480 . . . . . . . . . . . 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 406 . . . . . . . . 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 1131 . . . . . . 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 405 . . . . 5 ((𝐺 Struct 𝑋𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
43423adant2 1129 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
44 structex 17087 . . . . . . 7 (𝐺 Struct 𝑋𝐺 ∈ V)
45 structn0fun 17088 . . . . . . 7 (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
4644, 45jca 510 . . . . . 6 (𝐺 Struct 𝑋 → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
47463ad2ant1 1131 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
48 simp3 1136 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
49 simp2 1135 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐸𝑉)
50 setsfun0 17109 . . . . 5 (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
5147, 48, 49, 50syl12anc 833 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52443ad2ant1 1131 . . . . . 6 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐺 ∈ V)
53 setsdm 17107 . . . . . 6 ((𝐺 ∈ V ∧ 𝐸𝑉) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
5452, 49, 53syl2anc 582 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
55 fveq2 6890 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = (...‘⟨(1st𝑋), (2nd𝑋)⟩))
56 df-ov 7414 . . . . . . . . . . . . . . . . 17 ((1st𝑋)...(2nd𝑋)) = (...‘⟨(1st𝑋), (2nd𝑋)⟩)
5755, 56eqtr4di 2788 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = ((1st𝑋)...(2nd𝑋)))
5857sseq2d 4013 . . . . . . . . . . . . . . 15 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋))))
5958adantr 479 . . . . . . . . . . . . . 14 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋))))
60 df-3an 1087 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) ↔ (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ))
61 nnz 12583 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℤ)
62 nnz 12583 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
63 nnz 12583 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
6461, 62, 633anim123i 1149 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
65 ssfzunsnext 13550 . . . . . . . . . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . . . . . . . . . 21 (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋))...if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)) = (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
6765, 66sseqtrdi 4031 . . . . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 405 . . . . . . . . 9 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
79783adant2 1129 . . . . . . . 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 405 . . . . . 6 ((𝐺 Struct 𝑋𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
82813adant2 1129 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
8354, 82eqsstrd 4019 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
84 isstruct2 17086 . . . 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 1341 . . 3 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
8685adantr 479 . 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 5151 . . 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 480 . 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 256 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 394  w3a 1085   = wceq 1539  wcel 2104  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  ifcif 4527  {csn 4627  cop 4633   class class class wbr 5147   × cxp 5673  dom cdm 5675  Fun wfun 6536  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  cr 11111  cle 11253  cn 12216  cz 12562  ...cfz 13488   Struct cstr 17083   sSet csts 17100
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-struct 17084  df-sets 17101
This theorem is referenced by:  setsexstruct2  17112  setsstruct  17113
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