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Theorem trcfilu 23049
Description: Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
trcfilu ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))))

Proof of Theorem trcfilu
Dummy variables 𝑎 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2 simp2l 1200 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (CauFilu𝑈))
3 iscfilu 23043 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
43biimpa 480 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
51, 2, 4syl2anc 587 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
65simpld 498 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (fBas‘𝑋))
7 simp3 1139 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐴𝑋)
8 simp2r 1201 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ¬ ∅ ∈ (𝐹t 𝐴))
9 trfbas2 22597 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹t 𝐴)))
109biimpar 481 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
116, 7, 8, 10syl21anc 837 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
122ad5antr 734 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐹 ∈ (CauFilu𝑈))
131adantr 484 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
1413elfvexd 6711 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑋 ∈ V)
157adantr 484 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴𝑋)
1614, 15ssexd 5193 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴 ∈ V)
1716ad4antr 732 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐴 ∈ V)
18 simplr 769 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑎𝐹)
19 elrestr 16808 . . . . . . 7 ((𝐹 ∈ (CauFilu𝑈) ∧ 𝐴 ∈ V ∧ 𝑎𝐹) → (𝑎𝐴) ∈ (𝐹t 𝐴))
2012, 17, 18, 19syl3anc 1372 . . . . . 6 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎𝐴) ∈ (𝐹t 𝐴))
21 inxp 5676 . . . . . . 7 ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎𝐴) × (𝑎𝐴))
22 simpr 488 . . . . . . . . 9 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
2322ssrind 4127 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
24 simpllr 776 . . . . . . . 8 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
2523, 24sseqtrrd 3919 . . . . . . 7 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ 𝑤)
2621, 25eqsstrrid 3927 . . . . . 6 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤)
27 id 22 . . . . . . . . 9 (𝑏 = (𝑎𝐴) → 𝑏 = (𝑎𝐴))
2827sqxpeqd 5558 . . . . . . . 8 (𝑏 = (𝑎𝐴) → (𝑏 × 𝑏) = ((𝑎𝐴) × (𝑎𝐴)))
2928sseq1d 3909 . . . . . . 7 (𝑏 = (𝑎𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤))
3029rspcev 3527 . . . . . 6 (((𝑎𝐴) ∈ (𝐹t 𝐴) ∧ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
3120, 26, 30syl2anc 587 . . . . 5 (((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
325simprd 499 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3332r19.21bi 3122 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3433ad4ant13 751 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3531, 34r19.29a 3200 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
3616, 16xpexd 7495 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
37 simpr 488 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
38 elrest 16807 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
3938biimpa 480 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4013, 36, 37, 39syl21anc 837 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4135, 40r19.29a 3200 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
4241ralrimiva 3097 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
43 trust 22984 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
441, 7, 43syl2anc 587 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
45 iscfilu 23043 . . 3 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → ((𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ ((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)))
4644, 45syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ ((𝐹t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)))
4711, 42, 46mpbir2and 713 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055  Vcvv 3399  cin 3843  wss 3844  c0 4212   × cxp 5524  cfv 6340  (class class class)co 7173  t crest 16800  fBascfbas 20208  UnifOncust 22954  CauFiluccfilu 23041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-1st 7717  df-2nd 7718  df-rest 16802  df-fbas 20217  df-ust 22955  df-cfilu 23042
This theorem is referenced by:  ucnextcn  23059
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