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Theorem setsstruct2 16222
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 16194 . . . . . . 7 (𝐺 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)))
2 elin 3994 . . . . . . . . 9 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ (𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)))
3 elxp6 7435 . . . . . . . . . . 11 (𝑋 ∈ (ℕ × ℕ) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)))
4 eleq1 2866 . . . . . . . . . . . . 13 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
54adantr 473 . . . . . . . . . . . 12 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (𝑋 ∈ ≤ ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ))
6 simp3 1169 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
7 simp1l 1255 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (1st𝑋) ∈ ℕ)
86, 7ifcld 4322 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℕ)
98nnred 11329 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ∈ ℝ)
106nnred 11329 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → 𝐼 ∈ ℝ)
11 simp1r 1256 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (2nd𝑋) ∈ ℕ)
1211, 6ifcld 4322 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℕ)
1312nnred 11329 . . . . . . . . . . . . . . . . 17 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ∈ ℝ)
14 nnre 11320 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℝ)
1514adantr 473 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (1st𝑋) ∈ ℝ)
16 nnre 11320 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℝ)
1715, 16anim12i 607 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
18173adant2 1162 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
1918ancomd 454 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (1st𝑋) ∈ ℝ))
20 min1 12269 . . . . . . . . . . . . . . . . . 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 11320 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℝ)
2322adantl 474 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (2nd𝑋) ∈ ℝ)
2423, 16anim12i 607 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → ((2nd𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
25243adant2 1162 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ((2nd𝑋) ∈ ℝ ∧ 𝐼 ∈ ℝ))
2625ancomd 454 . . . . . . . . . . . . . . . . . 18 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → (𝐼 ∈ ℝ ∧ (2nd𝑋) ∈ ℝ))
27 max1 12265 . . . . . . . . . . . . . . . . . 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 10484 . . . . . . . . . . . . . . . 16 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼))
30 df-br 4844 . . . . . . . . . . . . . . . 16 (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)) ≤ if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼) ↔ ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ≤ )
3129, 30sylib 210 . . . . . . . . . . . . . . 15 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ≤ )
328, 12opelxpd 5350 . . . . . . . . . . . . . . 15 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ (ℕ × ℕ))
3331, 32elind 3996 . . . . . . . . . . . . . 14 ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ ∧ 𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
34333exp 1149 . . . . . . . . . . . . 13 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
3534adantl 474 . . . . . . . . . . . 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 232 . . . . . . . . . . 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 209 . . . . . . . . . 10 (𝑋 ∈ (ℕ × ℕ) → (𝑋 ∈ ≤ → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))))
3837impcom 397 . . . . . . . . 9 ((𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
392, 38sylbi 209 . . . . . . . 8 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
40393ad2ant1 1164 . . . . . . 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 209 . . . . . 6 (𝐺 Struct 𝑋 → (𝐼 ∈ ℕ → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ))))
4241imp 396 . . . . 5 ((𝐺 Struct 𝑋𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
43423adant2 1162 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
44 structex 16195 . . . . . . 7 (𝐺 Struct 𝑋𝐺 ∈ V)
45 structn0fun 16196 . . . . . . 7 (𝐺 Struct 𝑋 → Fun (𝐺 ∖ {∅}))
4644, 45jca 508 . . . . . 6 (𝐺 Struct 𝑋 → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
47463ad2ant1 1164 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})))
48 simp3 1169 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐼 ∈ ℕ)
49 simp2 1168 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐸𝑉)
50 setsfun0 16220 . . . . 5 (((𝐺 ∈ V ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼 ∈ ℕ ∧ 𝐸𝑉)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
5147, 48, 49, 50syl12anc 866 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))
52443ad2ant1 1164 . . . . . . 7 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → 𝐺 ∈ V)
5352, 49jca 508 . . . . . 6 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 ∈ V ∧ 𝐸𝑉))
54 setsdm 16218 . . . . . 6 ((𝐺 ∈ V ∧ 𝐸𝑉) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
5553, 54syl 17 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) = (dom 𝐺 ∪ {𝐼}))
56 fveq2 6411 . . . . . . . . . . . . . . . . 17 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = (...‘⟨(1st𝑋), (2nd𝑋)⟩))
57 df-ov 6881 . . . . . . . . . . . . . . . . 17 ((1st𝑋)...(2nd𝑋)) = (...‘⟨(1st𝑋), (2nd𝑋)⟩)
5856, 57syl6eqr 2851 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (...‘𝑋) = ((1st𝑋)...(2nd𝑋)))
5958sseq2d 3829 . . . . . . . . . . . . . . 15 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋))))
6059adantr 473 . . . . . . . . . . . . . 14 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) ↔ dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋))))
61 df-3an 1110 . . . . . . . . . . . . . . . . . 18 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) ↔ (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ))
62 nnz 11689 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑋) ∈ ℕ → (1st𝑋) ∈ ℤ)
63 nnz 11689 . . . . . . . . . . . . . . . . . . . . 21 ((2nd𝑋) ∈ ℕ → (2nd𝑋) ∈ ℤ)
64 nnz 11689 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ ℕ → 𝐼 ∈ ℤ)
6562, 63, 643anim123i 1191 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ))
66 ssfzunsnext 12640 . . . . . . . . . . . . . . . . . . . . 21 ((dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) ∧ ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋))...if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)))
67 df-ov 6881 . . . . . . . . . . . . . . . . . . . . 21 (if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋))...if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)) = (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
6866, 67syl6sseq 3847 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) ∧ ((1st𝑋) ∈ ℤ ∧ (2nd𝑋) ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
6965, 68sylan2 587 . . . . . . . . . . . . . . . . . . 19 ((dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) ∧ ((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ)) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
7069ex 402 . . . . . . . . . . . . . . . . . 18 (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
7161, 70syl5bir 235 . . . . . . . . . . . . . . . . 17 (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → ((((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) ∧ 𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
7271expd 405 . . . . . . . . . . . . . . . 16 (dom 𝐺 ⊆ ((1st𝑋)...(2nd𝑋)) → (((1st𝑋) ∈ ℕ ∧ (2nd𝑋) ∈ ℕ) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7372com12 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)⟩))))
7473adantl 474 . . . . . . . . . . . . . 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)⟩))))
7560, 74sylbid 232 . . . . . . . . . . . . 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)⟩))))
763, 75sylbi 209 . . . . . . . . . . . 12 (𝑋 ∈ (ℕ × ℕ) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7776adantl 474 . . . . . . . . . . 11 ((𝑋 ∈ ≤ ∧ 𝑋 ∈ (ℕ × ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
782, 77sylbi 209 . . . . . . . . . 10 (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → (dom 𝐺 ⊆ (...‘𝑋) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))))
7978imp 396 . . . . . . . . 9 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
80793adant2 1162 . . . . . . . 8 ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (...‘𝑋)) → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
811, 80sylbi 209 . . . . . . 7 (𝐺 Struct 𝑋 → (𝐼 ∈ ℕ → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)))
8281imp 396 . . . . . 6 ((𝐺 Struct 𝑋𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
83823adant2 1162 . . . . 5 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (dom 𝐺 ∪ {𝐼}) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
8455, 83eqsstrd 3835 . . . 4 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → dom (𝐺 sSet ⟨𝐼, 𝐸⟩) ⊆ (...‘⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩))
85 isstruct2 16194 . . . 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)⟩)))
8643, 51, 84, 85syl3anbrc 1444 . . 3 ((𝐺 Struct 𝑋𝐸𝑉𝐼 ∈ ℕ) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨if(𝐼 ≤ (1st𝑋), 𝐼, (1st𝑋)), if(𝐼 ≤ (2nd𝑋), (2nd𝑋), 𝐼)⟩)
8786adantr 473 . 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)⟩)
88 breq2 4847 . . 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)⟩))
8988adantl 474 . 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)⟩))
9087, 89mpbird 249 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385  cdif 3766  cun 3767  cin 3768  wss 3769  c0 4115  ifcif 4277  {csn 4368  cop 4374   class class class wbr 4843   × cxp 5310  dom cdm 5312  Fun wfun 6095  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  cr 10223  cle 10364  cn 11312  cz 11666  ...cfz 12580   Struct cstr 16180   sSet csts 16182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-struct 16186  df-sets 16191
This theorem is referenced by:  setsexstruct2  16223  setsstruct  16224
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