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Theorem trcfilu 23646
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 1136 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2 simp2l 1199 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (CauFilu𝑈))
3 iscfilu 23640 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
43biimpa 477 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
51, 2, 4syl2anc 584 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
65simpld 495 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐹 ∈ (fBas‘𝑋))
7 simp3 1138 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → 𝐴𝑋)
8 simp2r 1200 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ¬ ∅ ∈ (𝐹t 𝐴))
9 trfbas2 23194 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) → ((𝐹t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹t 𝐴)))
109biimpar 478 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝑋) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
116, 7, 8, 10syl21anc 836 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (fBas‘𝐴))
122ad5antr 732 . . . . . . 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 6881 . . . . . . . . 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 5281 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝐴 ∈ V)
1716ad4antr 730 . . . . . . 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 767 . . . . . . 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 17310 . . . . . . 7 ((𝐹 ∈ (CauFilu𝑈) ∧ 𝐴 ∈ V ∧ 𝑎𝐹) → (𝑎𝐴) ∈ (𝐹t 𝐴))
2012, 17, 18, 19syl3anc 1371 . . . . . 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 5788 . . . . . . 7 ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎𝐴) × (𝑎𝐴))
22 simpr 485 . . . . . . . . 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 4195 . . . . . . . 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 774 . . . . . . . 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 3985 . . . . . . 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 3993 . . . . . 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 5665 . . . . . . . 8 (𝑏 = (𝑎𝐴) → (𝑏 × 𝑏) = ((𝑎𝐴) × (𝑎𝐴)))
2928sseq1d 3975 . . . . . . 7 (𝑏 = (𝑎𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤))
3029rspcev 3581 . . . . . 6 (((𝑎𝐴) ∈ (𝐹t 𝐴) ∧ ((𝑎𝐴) × (𝑎𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
3120, 26, 30syl2anc 584 . . . . 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 496 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3332r19.21bi 3234 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3433ad4ant13 749 . . . . 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 3159 . . . 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 7685 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V)
37 simpr 485 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈t (𝐴 × 𝐴)))
38 elrest 17309 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈t (𝐴 × 𝐴)) ↔ ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))))
3938biimpa 477 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4013, 36, 37, 39syl21anc 836 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑣𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))
4135, 40r19.29a 3159 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) ∧ 𝑤 ∈ (𝑈t (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
4241ralrimiva 3143 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → ∀𝑤 ∈ (𝑈t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤)
43 trust 23581 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
441, 7, 43syl2anc 584 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
45 iscfilu 23640 . . 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 711 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  c0 4282   × cxp 5631  cfv 6496  (class class class)co 7357  t crest 17302  fBascfbas 20784  UnifOncust 23551  CauFiluccfilu 23638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-rest 17304  df-fbas 20793  df-ust 23552  df-cfilu 23639
This theorem is referenced by:  ucnextcn  23656
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