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Theorem trcfilu 22008
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 1059 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2 simp2l 1085 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (CauFilu𝑈))
3 iscfilu 22002 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
43biimpa 501 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
51, 2, 4syl2anc 692 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
65simpld 475 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (fBas‘𝑋))
7 simp3 1061 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐴𝑋)
8 simp2r 1086 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ¬ ∅ ∈ (𝐹t 𝐴))
9 trfbas2 21557 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹t 𝐴)))
109biimpar 502 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
116, 7, 8, 10syl21anc 1322 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
122ad5antr 769 . . . . . . 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 481 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋))
1413elfvexd 6179 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑋 ∈ V)
157adantr 481 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴𝑋)
1614, 15ssexd 4765 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴 ∈ V)
1716ad4antr 767 . . . . . . 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 791 . . . . . . 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 16010 . . . . . . 7 ((𝐹 ∈ (CauFilu𝑈) ∧ 𝐴 ∈ V ∧ 𝑎𝐹) → (𝑎𝐴) ∈ (𝐹t 𝐴))
2012, 17, 18, 19syl3anc 1323 . . . . . 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 5214 . . . . . . 7 ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎𝐴) × (𝑎𝐴))
22 simpr 477 . . . . . . . . 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)
23 ssrin 3816 . . . . . . . . 9 ((𝑎 × 𝑎) ⊆ 𝑣 → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
2422, 23syl 17 . . . . . . . 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)))
25 simpllr 798 . . . . . . . 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)))
2624, 25sseqtr4d 3621 . . . . . . 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)
2721, 26syl5eqssr 3629 . . . . . 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)
28 id 22 . . . . . . . . 9 (𝑏 = (𝑎𝐴) → 𝑏 = (𝑎𝐴))
2928sqxpeqd 5101 . . . . . . . 8 (𝑏 = (𝑎𝐴) → (𝑏 × 𝑏) = ((𝑎𝐴) × (𝑎𝐴)))
3029sseq1d 3611 . . . . . . 7 (𝑏 = (𝑎𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤))
3130rspcev 3295 . . . . . 6 (((𝑎𝐴) ∈ (𝐹t 𝐴) ∧ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
3220, 27, 31syl2anc 692 . . . . 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)
335simprd 479 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3433r19.21bi 2927 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
35343ad2antr2 1225 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ (𝑤 ∈ (𝑈t (𝐴 × 𝐴)) ∧ 𝑣𝑈𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
36353anassrs 1287 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3732, 36r19.29a 3071 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) ∧ 𝑣𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
38 xpexg 6913 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
3916, 16, 38syl2anc 692 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
40 simpr 477 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
41 elrest 16009 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
4241biimpa 501 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4313, 39, 40, 42syl21anc 1322 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4437, 43r19.29a 3071 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
4544ralrimiva 2960 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
46 trust 21943 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
471, 7, 46syl2anc 692 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
48 iscfilu 22002 . . 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)))
4947, 48syl 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)))
5011, 45, 49mpbir2and 956 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555  c0 3891   × cxp 5072  cfv 5847  (class class class)co 6604  t crest 16002  fBascfbas 19653  UnifOncust 21913  CauFiluccfilu 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-rest 16004  df-fbas 19662  df-ust 21914  df-cfilu 22001
This theorem is referenced by:  ucnextcn  22018
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