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Theorem trust 22080
Description: The trace of a uniform structure 𝑈 on a subset 𝐴 is a uniform structure on 𝐴. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
Assertion
Ref Expression
trust ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))

Proof of Theorem trust
Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 restsspw 16139 . . . 4 (𝑈t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴)
21a1i 11 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴))
3 inxp 5287 . . . . . 6 ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) = ((𝑋𝐴) × (𝑋𝐴))
4 sseqin2 3850 . . . . . . . 8 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
54biimpi 206 . . . . . . 7 (𝐴𝑋 → (𝑋𝐴) = 𝐴)
65sqxpeqd 5175 . . . . . 6 (𝐴𝑋 → ((𝑋𝐴) × (𝑋𝐴)) = (𝐴 × 𝐴))
73, 6syl5eq 2697 . . . . 5 (𝐴𝑋 → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
87adantl 481 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9 simpl 472 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
10 elfvex 6259 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
1110adantr 480 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
12 simpr 476 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
1311, 12ssexd 4838 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
14 xpexg 7002 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
1513, 13, 14syl2anc 694 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 × 𝐴) ∈ V)
16 ustbasel 22057 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
1716adantr 480 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
18 elrestr 16136 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ (𝑋 × 𝑋) ∈ 𝑈) → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
199, 15, 17, 18syl3anc 1366 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((𝑋 × 𝑋) ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
208, 19eqeltrrd 2731 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 × 𝐴) ∈ (𝑈t (𝐴 × 𝐴)))
219ad5antr 773 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
2215ad5antr 773 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
23 simplr 807 . . . . . . . . . . 11 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢𝑈)
24 simp-4r 824 . . . . . . . . . . . . . 14 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ∈ 𝒫 (𝐴 × 𝐴))
2524elpwid 4203 . . . . . . . . . . . . 13 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ⊆ (𝐴 × 𝐴))
2612ad5antr 773 . . . . . . . . . . . . . 14 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝐴𝑋)
27 xpss12 5158 . . . . . . . . . . . . . 14 ((𝐴𝑋𝐴𝑋) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
2826, 26, 27syl2anc 694 . . . . . . . . . . . . 13 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ⊆ (𝑋 × 𝑋))
2925, 28sstrd 3646 . . . . . . . . . . . 12 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ⊆ (𝑋 × 𝑋))
30 ustssxp 22055 . . . . . . . . . . . . 13 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
3121, 23, 30syl2anc 694 . . . . . . . . . . . 12 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢 ⊆ (𝑋 × 𝑋))
3229, 31unssd 3822 . . . . . . . . . . 11 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤𝑢) ⊆ (𝑋 × 𝑋))
33 ssun2 3810 . . . . . . . . . . . 12 𝑢 ⊆ (𝑤𝑢)
34 ustssel 22056 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈 ∧ (𝑤𝑢) ⊆ (𝑋 × 𝑋)) → (𝑢 ⊆ (𝑤𝑢) → (𝑤𝑢) ∈ 𝑈))
3533, 34mpi 20 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈 ∧ (𝑤𝑢) ⊆ (𝑋 × 𝑋)) → (𝑤𝑢) ∈ 𝑈)
3621, 23, 32, 35syl3anc 1366 . . . . . . . . . 10 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤𝑢) ∈ 𝑈)
37 df-ss 3621 . . . . . . . . . . . . . 14 (𝑤 ⊆ (𝐴 × 𝐴) ↔ (𝑤 ∩ (𝐴 × 𝐴)) = 𝑤)
3825, 37sylib 208 . . . . . . . . . . . . 13 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤 ∩ (𝐴 × 𝐴)) = 𝑤)
3938uneq1d 3799 . . . . . . . . . . . 12 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ((𝑤 ∩ (𝐴 × 𝐴)) ∪ (𝑢 ∩ (𝐴 × 𝐴))) = (𝑤 ∪ (𝑢 ∩ (𝐴 × 𝐴))))
40 simpr 476 . . . . . . . . . . . . . 14 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
41 simpllr 815 . . . . . . . . . . . . . 14 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣𝑤)
4240, 41eqsstr3d 3673 . . . . . . . . . . . . 13 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑤)
43 ssequn2 3819 . . . . . . . . . . . . 13 ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑤 ↔ (𝑤 ∪ (𝑢 ∩ (𝐴 × 𝐴))) = 𝑤)
4442, 43sylib 208 . . . . . . . . . . . 12 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝑤 ∪ (𝑢 ∩ (𝐴 × 𝐴))) = 𝑤)
4539, 44eqtr2d 2686 . . . . . . . . . . 11 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 = ((𝑤 ∩ (𝐴 × 𝐴)) ∪ (𝑢 ∩ (𝐴 × 𝐴))))
46 indir 3908 . . . . . . . . . . 11 ((𝑤𝑢) ∩ (𝐴 × 𝐴)) = ((𝑤 ∩ (𝐴 × 𝐴)) ∪ (𝑢 ∩ (𝐴 × 𝐴)))
4745, 46syl6eqr 2703 . . . . . . . . . 10 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 = ((𝑤𝑢) ∩ (𝐴 × 𝐴)))
48 ineq1 3840 . . . . . . . . . . . 12 (𝑥 = (𝑤𝑢) → (𝑥 ∩ (𝐴 × 𝐴)) = ((𝑤𝑢) ∩ (𝐴 × 𝐴)))
4948eqeq2d 2661 . . . . . . . . . . 11 (𝑥 = (𝑤𝑢) → (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) ↔ 𝑤 = ((𝑤𝑢) ∩ (𝐴 × 𝐴))))
5049rspcev 3340 . . . . . . . . . 10 (((𝑤𝑢) ∈ 𝑈𝑤 = ((𝑤𝑢) ∩ (𝐴 × 𝐴))) → ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
5136, 47, 50syl2anc 694 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
52 elrest 16135 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))))
5352biimpar 501 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
5421, 22, 51, 53syl21anc 1365 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
55 elrest 16135 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑢𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))))
5655biimpa 500 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑢𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
5715, 56syldanl 735 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑢𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
5857ad2antrr 762 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) → ∃𝑢𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
5954, 58r19.29a 3107 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) ∧ 𝑣𝑤) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
6059ex 449 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ 𝒫 (𝐴 × 𝐴)) → (𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))))
6160ralrimiva 2995 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → ∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))))
629ad5antr 773 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
6315ad5antr 773 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝐴 × 𝐴) ∈ V)
64 simpllr 815 . . . . . . . . . 10 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑢𝑈)
65 simplr 807 . . . . . . . . . 10 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑥𝑈)
66 ustincl 22058 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈𝑥𝑈) → (𝑢𝑥) ∈ 𝑈)
6762, 64, 65, 66syl3anc 1366 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑢𝑥) ∈ 𝑈)
68 simprl 809 . . . . . . . . . . 11 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
69 simprr 811 . . . . . . . . . . 11 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
7068, 69ineq12d 3848 . . . . . . . . . 10 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑣𝑤) = ((𝑢 ∩ (𝐴 × 𝐴)) ∩ (𝑥 ∩ (𝐴 × 𝐴))))
71 inindir 3864 . . . . . . . . . 10 ((𝑢𝑥) ∩ (𝐴 × 𝐴)) = ((𝑢 ∩ (𝐴 × 𝐴)) ∩ (𝑥 ∩ (𝐴 × 𝐴)))
7270, 71syl6eqr 2703 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑣𝑤) = ((𝑢𝑥) ∩ (𝐴 × 𝐴)))
73 ineq1 3840 . . . . . . . . . . 11 (𝑦 = (𝑢𝑥) → (𝑦 ∩ (𝐴 × 𝐴)) = ((𝑢𝑥) ∩ (𝐴 × 𝐴)))
7473eqeq2d 2661 . . . . . . . . . 10 (𝑦 = (𝑢𝑥) → ((𝑣𝑤) = (𝑦 ∩ (𝐴 × 𝐴)) ↔ (𝑣𝑤) = ((𝑢𝑥) ∩ (𝐴 × 𝐴))))
7574rspcev 3340 . . . . . . . . 9 (((𝑢𝑥) ∈ 𝑈 ∧ (𝑣𝑤) = ((𝑢𝑥) ∩ (𝐴 × 𝐴))) → ∃𝑦𝑈 (𝑣𝑤) = (𝑦 ∩ (𝐴 × 𝐴)))
7667, 72, 75syl2anc 694 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → ∃𝑦𝑈 (𝑣𝑤) = (𝑦 ∩ (𝐴 × 𝐴)))
77 elrest 16135 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → ((𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑦𝑈 (𝑣𝑤) = (𝑦 ∩ (𝐴 × 𝐴))))
7877biimpar 501 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ ∃𝑦𝑈 (𝑣𝑤) = (𝑦 ∩ (𝐴 × 𝐴))) → (𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)))
7962, 63, 76, 78syl21anc 1365 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))) → (𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)))
8057adantr 480 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑢𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
819ad2antrr 762 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
8215ad2antrr 762 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
83 simpr 476 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
8452biimpa 500 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
8581, 82, 83, 84syl21anc 1365 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
86 reeanv 3136 . . . . . . . 8 (∃𝑢𝑈𝑥𝑈 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))) ↔ (∃𝑢𝑈 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ ∃𝑥𝑈 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))))
8780, 85, 86sylanbrc 699 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑢𝑈𝑥𝑈 (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) ∧ 𝑤 = (𝑥 ∩ (𝐴 × 𝐴))))
8879, 87r19.29vva 3110 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → (𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)))
8988ralrimiva 2995 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)))
90 simp-4l 823 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
91 simplr 807 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢𝑈)
92 ustdiag 22059 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → ( I ↾ 𝑋) ⊆ 𝑢)
9390, 91, 92syl2anc 694 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ( I ↾ 𝑋) ⊆ 𝑢)
94 simp-4r 824 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝐴𝑋)
95 inss1 3866 . . . . . . . . . . . . . 14 (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ ( I ↾ 𝑋)
96 resss 5457 . . . . . . . . . . . . . 14 ( I ↾ 𝑋) ⊆ I
9795, 96sstri 3645 . . . . . . . . . . . . 13 (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ I
98 iss 5482 . . . . . . . . . . . . 13 ((( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ I ↔ (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = ( I ↾ dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴))))
9997, 98mpbi 220 . . . . . . . . . . . 12 (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = ( I ↾ dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)))
100 simpr 476 . . . . . . . . . . . . . . . 16 ((𝐴𝑋𝑢𝐴) → 𝑢𝐴)
101 ssel2 3631 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝑢𝐴) → 𝑢𝑋)
102 equid 1985 . . . . . . . . . . . . . . . . . 18 𝑢 = 𝑢
103 resieq 5442 . . . . . . . . . . . . . . . . . 18 ((𝑢𝑋𝑢𝑋) → (𝑢( I ↾ 𝑋)𝑢𝑢 = 𝑢))
104102, 103mpbiri 248 . . . . . . . . . . . . . . . . 17 ((𝑢𝑋𝑢𝑋) → 𝑢( I ↾ 𝑋)𝑢)
105101, 101, 104syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝐴𝑋𝑢𝐴) → 𝑢( I ↾ 𝑋)𝑢)
106 breq2 4689 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑢 → (𝑢( I ↾ 𝑋)𝑣𝑢( I ↾ 𝑋)𝑢))
107106rspcev 3340 . . . . . . . . . . . . . . . 16 ((𝑢𝐴𝑢( I ↾ 𝑋)𝑢) → ∃𝑣𝐴 𝑢( I ↾ 𝑋)𝑣)
108100, 105, 107syl2anc 694 . . . . . . . . . . . . . . 15 ((𝐴𝑋𝑢𝐴) → ∃𝑣𝐴 𝑢( I ↾ 𝑋)𝑣)
109108ralrimiva 2995 . . . . . . . . . . . . . 14 (𝐴𝑋 → ∀𝑢𝐴𝑣𝐴 𝑢( I ↾ 𝑋)𝑣)
110 dminxp 5609 . . . . . . . . . . . . . 14 (dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑢𝐴𝑣𝐴 𝑢( I ↾ 𝑋)𝑣)
111109, 110sylibr 224 . . . . . . . . . . . . 13 (𝐴𝑋 → dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) = 𝐴)
112111reseq2d 5428 . . . . . . . . . . . 12 (𝐴𝑋 → ( I ↾ dom (( I ↾ 𝑋) ∩ (𝐴 × 𝐴))) = ( I ↾ 𝐴))
11399, 112syl5req 2698 . . . . . . . . . . 11 (𝐴𝑋 → ( I ↾ 𝐴) = (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)))
114113adantl 481 . . . . . . . . . 10 ((( I ↾ 𝑋) ⊆ 𝑢𝐴𝑋) → ( I ↾ 𝐴) = (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)))
115 ssrin 3871 . . . . . . . . . . 11 (( I ↾ 𝑋) ⊆ 𝑢 → (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
116115adantr 480 . . . . . . . . . 10 ((( I ↾ 𝑋) ⊆ 𝑢𝐴𝑋) → (( I ↾ 𝑋) ∩ (𝐴 × 𝐴)) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
117114, 116eqsstrd 3672 . . . . . . . . 9 ((( I ↾ 𝑋) ⊆ 𝑢𝐴𝑋) → ( I ↾ 𝐴) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
11893, 94, 117syl2anc 694 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ( I ↾ 𝐴) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
119 simpr 476 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
120118, 119sseqtr4d 3675 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ( I ↾ 𝐴) ⊆ 𝑣)
121120, 57r19.29a 3107 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → ( I ↾ 𝐴) ⊆ 𝑣)
12215ad3antrrr 766 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
123 ustinvel 22060 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢𝑈)
12490, 91, 123syl2anc 694 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑢𝑈)
125119cnveqd 5330 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
126 cnvin 5575 . . . . . . . . . . 11 (𝑢 ∩ (𝐴 × 𝐴)) = (𝑢(𝐴 × 𝐴))
127 cnvxp 5586 . . . . . . . . . . . 12 (𝐴 × 𝐴) = (𝐴 × 𝐴)
128127ineq2i 3844 . . . . . . . . . . 11 (𝑢(𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
129126, 128eqtri 2673 . . . . . . . . . 10 (𝑢 ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴))
130125, 129syl6eq 2701 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 = (𝑢 ∩ (𝐴 × 𝐴)))
131 ineq1 3840 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝑥 ∩ (𝐴 × 𝐴)) = (𝑢 ∩ (𝐴 × 𝐴)))
132131eqeq2d 2661 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑣 = (𝑥 ∩ (𝐴 × 𝐴)) ↔ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))))
133132rspcev 3340 . . . . . . . . 9 ((𝑢𝑈𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑥𝑈 𝑣 = (𝑥 ∩ (𝐴 × 𝐴)))
134124, 130, 133syl2anc 694 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑥𝑈 𝑣 = (𝑥 ∩ (𝐴 × 𝐴)))
135 elrest 16135 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑥𝑈 𝑣 = (𝑥 ∩ (𝐴 × 𝐴))))
136135biimpar 501 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ ∃𝑥𝑈 𝑣 = (𝑥 ∩ (𝐴 × 𝐴))) → 𝑣 ∈ (𝑈t (𝐴 × 𝐴)))
13790, 122, 134, 136syl21anc 1365 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → 𝑣 ∈ (𝑈t (𝐴 × 𝐴)))
138137, 57r19.29a 3107 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑣 ∈ (𝑈t (𝐴 × 𝐴)))
139 simp-4l 823 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → 𝑈 ∈ (UnifOn‘𝑋))
14015ad3antrrr 766 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → (𝐴 × 𝐴) ∈ V)
141 simplr 807 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → 𝑥𝑈)
142 elrestr 16136 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑥𝑈) → (𝑥 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
143139, 140, 141, 142syl3anc 1366 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → (𝑥 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
144 inss1 3866 . . . . . . . . . . . . . . 15 (𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥
145 coss1 5310 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥 → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥 ∘ (𝑥 ∩ (𝐴 × 𝐴))))
146 coss2 5311 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥 → (𝑥 ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥𝑥))
147145, 146sstrd 3646 . . . . . . . . . . . . . . 15 ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ 𝑥 → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥𝑥))
148144, 147ax-mp 5 . . . . . . . . . . . . . 14 ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥𝑥)
149 sstr 3644 . . . . . . . . . . . . . 14 ((((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑥𝑥) ∧ (𝑥𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
150148, 149mpan 706 . . . . . . . . . . . . 13 ((𝑥𝑥) ⊆ 𝑢 → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
151150adantl 481 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ 𝑢)
152 inss2 3867 . . . . . . . . . . . . . . 15 (𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
153 coss1 5310 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝑥 ∩ (𝐴 × 𝐴))))
154 coss2 5311 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)))
155153, 154sstrd 3646 . . . . . . . . . . . . . . 15 ((𝑥 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)))
156152, 155ax-mp 5 . . . . . . . . . . . . . 14 ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))
157 xpidtr 5553 . . . . . . . . . . . . . 14 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
158156, 157sstri 3645 . . . . . . . . . . . . 13 ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)
159158a1i 11 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))
160151, 159ssind 3870 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
161 id 22 . . . . . . . . . . . . . 14 (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) → 𝑤 = (𝑥 ∩ (𝐴 × 𝐴)))
162161, 161coeq12d 5319 . . . . . . . . . . . . 13 (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) → (𝑤𝑤) = ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))))
163162sseq1d 3665 . . . . . . . . . . . 12 (𝑤 = (𝑥 ∩ (𝐴 × 𝐴)) → ((𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)) ↔ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑢 ∩ (𝐴 × 𝐴))))
164163rspcev 3340 . . . . . . . . . . 11 (((𝑥 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ((𝑥 ∩ (𝐴 × 𝐴)) ∘ (𝑥 ∩ (𝐴 × 𝐴))) ⊆ (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
165143, 160, 164syl2anc 694 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) ∧ 𝑥𝑈) ∧ (𝑥𝑥) ⊆ 𝑢) → ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
166 ustexhalf 22061 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑢)
167166adantlr 751 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) → ∃𝑥𝑈 (𝑥𝑥) ⊆ 𝑢)
168165, 167r19.29a 3107 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢𝑈) → ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
169168ad4ant13 1315 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴)))
170119sseq2d 3666 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴))))
171170rexbidv 3081 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → (∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ (𝑢 ∩ (𝐴 × 𝐴))))
172169, 171mpbird 247 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑢𝑈) ∧ 𝑣 = (𝑢 ∩ (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣)
173172, 57r19.29a 3107 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣)
174121, 138, 1733jca 1261 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → (( I ↾ 𝐴) ⊆ 𝑣𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣))
17561, 89, 1743jca 1261 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑣 ∈ (𝑈t (𝐴 × 𝐴))) → (∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣)))
176175ralrimiva 2995 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ∀𝑣 ∈ (𝑈t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣)))
1772, 20, 1763jca 1261 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((𝑈t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∀𝑣 ∈ (𝑈t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣))))
178 isust 22054 . . 3 (𝐴 ∈ V → ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ↔ ((𝑈t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∀𝑣 ∈ (𝑈t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣)))))
17913, 178syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ↔ ((𝑈t (𝐴 × 𝐴)) ⊆ 𝒫 (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∀𝑣 ∈ (𝑈t (𝐴 × 𝐴))(∀𝑤 ∈ 𝒫 (𝐴 × 𝐴)(𝑣𝑤𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑣𝑤) ∈ (𝑈t (𝐴 × 𝐴)) ∧ (( I ↾ 𝐴) ⊆ 𝑣𝑣 ∈ (𝑈t (𝐴 × 𝐴)) ∧ ∃𝑤 ∈ (𝑈t (𝐴 × 𝐴))(𝑤𝑤) ⊆ 𝑣)))))
180177, 179mpbird 247 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191   class class class wbr 4685   I cid 5052   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  ccom 5147  cfv 5926  (class class class)co 6690  t crest 16128  UnifOncust 22050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-rest 16130  df-ust 22051
This theorem is referenced by:  restutop  22088  restutopopn  22089  ressust  22115  ressusp  22116  trcfilu  22145  cfiluweak  22146
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