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Theorem fmucnd 24317
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 24314 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈)) → 𝐶 ∈ (fBas‘𝑋))
41, 2, 3syl2anc 584 . . 3 (𝜑𝐶 ∈ (fBas‘𝑋))
5 fmucnd.2 . . . 4 (𝜑𝑉 ∈ (UnifOn‘𝑌))
6 fmucnd.3 . . . 4 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
7 isucn 24303 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
87simprbda 498 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) ∧ 𝐹 ∈ (𝑈 Cnu𝑉)) → 𝐹:𝑋𝑌)
91, 5, 6, 8syl21anc 838 . . 3 (𝜑𝐹:𝑋𝑌)
105elfvexd 6946 . . 3 (𝜑𝑌 ∈ V)
11 fmucnd.5 . . . 4 𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))
1211fbasrn 23908 . . 3 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌 ∈ V) → 𝐷 ∈ (fBas‘𝑌))
134, 9, 10, 12syl3anc 1370 . 2 (𝜑𝐷 ∈ (fBas‘𝑌))
14 simplr 769 . . . . . . . 8 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝐶)
15 eqid 2735 . . . . . . . 8 (𝐹𝑎) = (𝐹𝑎)
16 imaeq2 6076 . . . . . . . . 9 (𝑐 = 𝑎 → (𝐹𝑐) = (𝐹𝑎))
1716rspceeqv 3645 . . . . . . . 8 ((𝑎𝐶 ∧ (𝐹𝑎) = (𝐹𝑎)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
1814, 15, 17sylancl 586 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
19 imaexg 7936 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cnu𝑉) → (𝐹𝑎) ∈ V)
20 eqid 2735 . . . . . . . . . 10 (𝑐𝐶 ↦ (𝐹𝑐)) = (𝑐𝐶 ↦ (𝐹𝑐))
2120elrnmpt 5972 . . . . . . . . 9 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
226, 19, 213syl 18 . . . . . . . 8 (𝜑 → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2322ad3antrrr 730 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2418, 23mpbird 257 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)))
25 imaeq2 6076 . . . . . . . . 9 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
2625cbvmptv 5261 . . . . . . . 8 (𝑎𝐶 ↦ (𝐹𝑎)) = (𝑐𝐶 ↦ (𝐹𝑐))
2726rneqi 5951 . . . . . . 7 ran (𝑎𝐶 ↦ (𝐹𝑎)) = ran (𝑐𝐶 ↦ (𝐹𝑐))
2811, 27eqtri 2763 . . . . . 6 𝐷 = ran (𝑐𝐶 ↦ (𝐹𝑐))
2924, 28eleqtrrdi 2850 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ 𝐷)
309ffnd 6738 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
3130ad3antrrr 730 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝐹 Fn 𝑋)
32 fbelss 23857 . . . . . . . . 9 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑎𝐶) → 𝑎𝑋)
334, 32sylan 580 . . . . . . . 8 ((𝜑𝑎𝐶) → 𝑎𝑋)
3433ad4ant13 751 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝑋)
35 fmucndlem 24316 . . . . . . 7 ((𝐹 Fn 𝑋𝑎𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
3631, 34, 35syl2anc 584 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
37 eqid 2735 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
3837mpofun 7557 . . . . . . . 8 Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
39 funimass2 6651 . . . . . . . 8 ((Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4038, 39mpan 690 . . . . . . 7 ((𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4140adantl 481 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4236, 41eqsstrrd 4035 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣)
43 id 22 . . . . . . . 8 (𝑏 = (𝐹𝑎) → 𝑏 = (𝐹𝑎))
4443sqxpeqd 5721 . . . . . . 7 (𝑏 = (𝐹𝑎) → (𝑏 × 𝑏) = ((𝐹𝑎) × (𝐹𝑎)))
4544sseq1d 4027 . . . . . 6 (𝑏 = (𝐹𝑎) → ((𝑏 × 𝑏) ⊆ 𝑣 ↔ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣))
4645rspcev 3622 . . . . 5 (((𝐹𝑎) ∈ 𝐷 ∧ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
4729, 42, 46syl2anc 584 . . . 4 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
481adantr 480 . . . . 5 ((𝜑𝑣𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
492adantr 480 . . . . 5 ((𝜑𝑣𝑉) → 𝐶 ∈ (CauFilu𝑈))
505adantr 480 . . . . . 6 ((𝜑𝑣𝑉) → 𝑉 ∈ (UnifOn‘𝑌))
516adantr 480 . . . . . 6 ((𝜑𝑣𝑉) → 𝐹 ∈ (𝑈 Cnu𝑉))
52 simpr 484 . . . . . 6 ((𝜑𝑣𝑉) → 𝑣𝑉)
53 nfcv 2903 . . . . . . 7 𝑠⟨(𝐹𝑥), (𝐹𝑦)⟩
54 nfcv 2903 . . . . . . 7 𝑡⟨(𝐹𝑥), (𝐹𝑦)⟩
55 nfcv 2903 . . . . . . 7 𝑥⟨(𝐹𝑠), (𝐹𝑡)⟩
56 nfcv 2903 . . . . . . 7 𝑦⟨(𝐹𝑠), (𝐹𝑡)⟩
57 simpl 482 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑥 = 𝑠)
5857fveq2d 6911 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑥) = (𝐹𝑠))
59 simpr 484 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑦 = 𝑡)
6059fveq2d 6911 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑦) = (𝐹𝑡))
6158, 60opeq12d 4886 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 𝑡) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6253, 54, 55, 56, 61cbvmpo 7527 . . . . . 6 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑠𝑋, 𝑡𝑋 ↦ ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6348, 50, 51, 52, 62ucnprima 24307 . . . . 5 ((𝜑𝑣𝑉) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈)
64 cfiluexsm 24315 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈) ∧ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6548, 49, 63, 64syl3anc 1370 . . . 4 ((𝜑𝑣𝑉) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6647, 65r19.29a 3160 . . 3 ((𝜑𝑣𝑉) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
6766ralrimiva 3144 . 2 (𝜑 → ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
68 iscfilu 24313 . . 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 713 1 (𝜑𝐷 ∈ (CauFilu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  wss 3963  cop 4637   class class class wbr 5148  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  fBascfbas 21370  UnifOncust 24224   Cnucucn 24300  CauFiluccfilu 24311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-fbas 21379  df-ust 24225  df-ucn 24301  df-cfilu 24312
This theorem is referenced by:  ucnextcn  24329
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