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Theorem fmucnd 24291
Description: The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)
Hypotheses
Ref Expression
fmucnd.1 (𝜑𝑈 ∈ (UnifOn‘𝑋))
fmucnd.2 (𝜑𝑉 ∈ (UnifOn‘𝑌))
fmucnd.3 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
fmucnd.4 (𝜑𝐶 ∈ (CauFilu𝑈))
fmucnd.5 𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))
Assertion
Ref Expression
fmucnd (𝜑𝐷 ∈ (CauFilu𝑉))
Distinct variable groups:   𝐶,𝑎   𝐷,𝑎   𝐹,𝑎   𝑉,𝑎   𝑋,𝑎   𝑌,𝑎   𝜑,𝑎
Allowed substitution hint:   𝑈(𝑎)

Proof of Theorem fmucnd
Dummy variables 𝑐 𝑏 𝑣 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmucnd.1 . . . 4 (𝜑𝑈 ∈ (UnifOn‘𝑋))
2 fmucnd.4 . . . 4 (𝜑𝐶 ∈ (CauFilu𝑈))
3 cfilufbas 24288 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈)) → 𝐶 ∈ (fBas‘𝑋))
41, 2, 3syl2anc 582 . . 3 (𝜑𝐶 ∈ (fBas‘𝑋))
5 fmucnd.2 . . . 4 (𝜑𝑉 ∈ (UnifOn‘𝑌))
6 fmucnd.3 . . . 4 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
7 isucn 24277 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
87simprbda 497 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) ∧ 𝐹 ∈ (𝑈 Cnu𝑉)) → 𝐹:𝑋𝑌)
91, 5, 6, 8syl21anc 836 . . 3 (𝜑𝐹:𝑋𝑌)
105elfvexd 6942 . . 3 (𝜑𝑌 ∈ V)
11 fmucnd.5 . . . 4 𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))
1211fbasrn 23882 . . 3 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌 ∈ V) → 𝐷 ∈ (fBas‘𝑌))
134, 9, 10, 12syl3anc 1368 . 2 (𝜑𝐷 ∈ (fBas‘𝑌))
14 simplr 767 . . . . . . . 8 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝐶)
15 eqid 2726 . . . . . . . 8 (𝐹𝑎) = (𝐹𝑎)
16 imaeq2 6067 . . . . . . . . 9 (𝑐 = 𝑎 → (𝐹𝑐) = (𝐹𝑎))
1716rspceeqv 3630 . . . . . . . 8 ((𝑎𝐶 ∧ (𝐹𝑎) = (𝐹𝑎)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
1814, 15, 17sylancl 584 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
19 imaexg 7928 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cnu𝑉) → (𝐹𝑎) ∈ V)
20 eqid 2726 . . . . . . . . . 10 (𝑐𝐶 ↦ (𝐹𝑐)) = (𝑐𝐶 ↦ (𝐹𝑐))
2120elrnmpt 5964 . . . . . . . . 9 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
226, 19, 213syl 18 . . . . . . . 8 (𝜑 → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2322ad3antrrr 728 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2418, 23mpbird 256 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)))
25 imaeq2 6067 . . . . . . . . 9 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
2625cbvmptv 5268 . . . . . . . 8 (𝑎𝐶 ↦ (𝐹𝑎)) = (𝑐𝐶 ↦ (𝐹𝑐))
2726rneqi 5945 . . . . . . 7 ran (𝑎𝐶 ↦ (𝐹𝑎)) = ran (𝑐𝐶 ↦ (𝐹𝑐))
2811, 27eqtri 2754 . . . . . 6 𝐷 = ran (𝑐𝐶 ↦ (𝐹𝑐))
2924, 28eleqtrrdi 2837 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ 𝐷)
309ffnd 6731 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
3130ad3antrrr 728 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝐹 Fn 𝑋)
32 fbelss 23831 . . . . . . . . 9 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑎𝐶) → 𝑎𝑋)
334, 32sylan 578 . . . . . . . 8 ((𝜑𝑎𝐶) → 𝑎𝑋)
3433ad4ant13 749 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝑋)
35 fmucndlem 24290 . . . . . . 7 ((𝐹 Fn 𝑋𝑎𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
3631, 34, 35syl2anc 582 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
37 eqid 2726 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
3837mpofun 7551 . . . . . . . 8 Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
39 funimass2 6644 . . . . . . . 8 ((Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4038, 39mpan 688 . . . . . . 7 ((𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4140adantl 480 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4236, 41eqsstrrd 4019 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣)
43 id 22 . . . . . . . 8 (𝑏 = (𝐹𝑎) → 𝑏 = (𝐹𝑎))
4443sqxpeqd 5716 . . . . . . 7 (𝑏 = (𝐹𝑎) → (𝑏 × 𝑏) = ((𝐹𝑎) × (𝐹𝑎)))
4544sseq1d 4011 . . . . . 6 (𝑏 = (𝐹𝑎) → ((𝑏 × 𝑏) ⊆ 𝑣 ↔ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣))
4645rspcev 3608 . . . . 5 (((𝐹𝑎) ∈ 𝐷 ∧ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
4729, 42, 46syl2anc 582 . . . 4 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
481adantr 479 . . . . 5 ((𝜑𝑣𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
492adantr 479 . . . . 5 ((𝜑𝑣𝑉) → 𝐶 ∈ (CauFilu𝑈))
505adantr 479 . . . . . 6 ((𝜑𝑣𝑉) → 𝑉 ∈ (UnifOn‘𝑌))
516adantr 479 . . . . . 6 ((𝜑𝑣𝑉) → 𝐹 ∈ (𝑈 Cnu𝑉))
52 simpr 483 . . . . . 6 ((𝜑𝑣𝑉) → 𝑣𝑉)
53 nfcv 2892 . . . . . . 7 𝑠⟨(𝐹𝑥), (𝐹𝑦)⟩
54 nfcv 2892 . . . . . . 7 𝑡⟨(𝐹𝑥), (𝐹𝑦)⟩
55 nfcv 2892 . . . . . . 7 𝑥⟨(𝐹𝑠), (𝐹𝑡)⟩
56 nfcv 2892 . . . . . . 7 𝑦⟨(𝐹𝑠), (𝐹𝑡)⟩
57 simpl 481 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑥 = 𝑠)
5857fveq2d 6907 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑥) = (𝐹𝑠))
59 simpr 483 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑦 = 𝑡)
6059fveq2d 6907 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑦) = (𝐹𝑡))
6158, 60opeq12d 4889 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 𝑡) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6253, 54, 55, 56, 61cbvmpo 7521 . . . . . 6 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑠𝑋, 𝑡𝑋 ↦ ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6348, 50, 51, 52, 62ucnprima 24281 . . . . 5 ((𝜑𝑣𝑉) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈)
64 cfiluexsm 24289 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈) ∧ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6548, 49, 63, 64syl3anc 1368 . . . 4 ((𝜑𝑣𝑉) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6647, 65r19.29a 3152 . . 3 ((𝜑𝑣𝑉) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
6766ralrimiva 3136 . 2 (𝜑 → ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
68 iscfilu 24287 . . 3 (𝑉 ∈ (UnifOn‘𝑌) → (𝐷 ∈ (CauFilu𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
695, 68syl 17 . 2 (𝜑 → (𝐷 ∈ (CauFilu𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
7013, 67, 69mpbir2and 711 1 (𝜑𝐷 ∈ (CauFilu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  wrex 3060  Vcvv 3462  wss 3947  cop 4639   class class class wbr 5155  cmpt 5238   × cxp 5682  ccnv 5683  ran crn 5685  cima 5687  Fun wfun 6550   Fn wfn 6551  wf 6552  cfv 6556  (class class class)co 7426  cmpo 7428  fBascfbas 21333  UnifOncust 24198   Cnucucn 24274  CauFiluccfilu 24285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8005  df-2nd 8006  df-map 8859  df-fbas 21342  df-ust 24199  df-ucn 24275  df-cfilu 24286
This theorem is referenced by:  ucnextcn  24303
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