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Theorem setsstruct2 17134
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 17109 . . . . . . 7 (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2 elin 3960 . . . . . . . . 9 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3 elxp6 8021 . . . . . . . . . . 11 (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)))
4 eleq1 2816 . . . . . . . . . . . . 13 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
54adantr 480 . . . . . . . . . . . 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 4570 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℕ)
98nnred 12249 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℝ)
106nnred 12249 . . . . . . . . . . . . . . . . 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 4570 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℕ)
1312nnred 12249 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℝ)
14 nnre 12241 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℝ)
1514adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℝ)
16 nnre 12241 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℝ)
1715, 16anim12i 612 . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (1st𝑋) ∈ ℝ))
20 min1 13192 . . . . . . . . . . . . . . . . . 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 12241 . . . . . . . . . . . . . . . . . . . . . 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 1129 . . . . . . . . . . . . . . . . . . 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 13188 . . . . . . . . . . . . . . . . . 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 11393 . . . . . . . . . . . . . . . 16 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼))
30 df-br 5143 . . . . . . . . . . . . . . . 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 5711 . . . . . . . . . . . . . . 15 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ (ℕ × ℕ))
3331, 32elind 4190 . . . . . . . . . . . . . 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 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 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 406 . . . . 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 17110 . . . . . . 7 (𝐺 Struct 𝑋𝐺 ∈ V)
45 structn0fun 17111 . . . . . . 7 (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
4644, 45jca 511 . . . . . 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 17132 . . . . 5 (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
5147, 48, 49, 50syl12anc 836 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52443ad2ant1 1131 . . . . . 6 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐺 ∈ V)
53 setsdm 17130 . . . . . 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 6891 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = (...‘⟨(1st𝑋), (2nd𝑋)⟩))
56 df-ov 7417 . . . . . . . . . . . . . . . . 17 ((1st𝑋)...(2nd𝑋)) = (...‘⟨(1st𝑋), (2nd𝑋)⟩)
5755, 56eqtr4di 2785 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = ((1st𝑋)...(2nd𝑋)))
5857sseq2d 4010 . . . . . . . . . . . . . . 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 1087 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) ↔ (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ))
61 nnz 12601 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℤ)
62 nnz 12601 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
63 nnz 12601 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
6461, 62, 633anim123i 1149 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
65 ssfzunsnext 13570 . . . . . . . . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . . . . 21 (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋))...if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)) = (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
6765, 66sseqtrdi 4028 . . . . . . . . . . . . . . . . . . . 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 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 406 . . . . . 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 4016 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
84 isstruct2 17109 . . . 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 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 5146 . . 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 1085   = wceq 1534  wcel 2099  Vcvv 3469  cdif 3941  cun 3942  cin 3943  wss 3944  c0 4318  ifcif 4524  {csn 4624  cop 4630   class class class wbr 5142   × cxp 5670  dom cdm 5672  Fun wfun 6536  cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  cr 11129  cle 11271  cn 12234  cz 12580  ...cfz 13508   Struct cstr 17106   sSet csts 17123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124
This theorem is referenced by:  setsexstruct2  17135  setsstruct  17136
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