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Theorem fmucnd 22594
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 22591 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈)) → 𝐶 ∈ (fBas‘𝑋))
41, 2, 3syl2anc 576 . . 3 (𝜑𝐶 ∈ (fBas‘𝑋))
5 fmucnd.2 . . . 4 (𝜑𝑉 ∈ (UnifOn‘𝑌))
6 fmucnd.3 . . . 4 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
7 isucn 22580 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
87simprbda 491 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) ∧ 𝐹 ∈ (𝑈 Cnu𝑉)) → 𝐹:𝑋𝑌)
91, 5, 6, 8syl21anc 825 . . 3 (𝜑𝐹:𝑋𝑌)
105elfvexd 6528 . . 3 (𝜑𝑌 ∈ V)
11 fmucnd.5 . . . 4 𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))
1211fbasrn 22186 . . 3 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌 ∈ V) → 𝐷 ∈ (fBas‘𝑌))
134, 9, 10, 12syl3anc 1351 . 2 (𝜑𝐷 ∈ (fBas‘𝑌))
14 simplr 756 . . . . . . . 8 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝐶)
15 eqid 2772 . . . . . . . 8 (𝐹𝑎) = (𝐹𝑎)
16 imaeq2 5760 . . . . . . . . 9 (𝑐 = 𝑎 → (𝐹𝑐) = (𝐹𝑎))
1716rspceeqv 3547 . . . . . . . 8 ((𝑎𝐶 ∧ (𝐹𝑎) = (𝐹𝑎)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
1814, 15, 17sylancl 577 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
19 imaexg 7429 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cnu𝑉) → (𝐹𝑎) ∈ V)
20 eqid 2772 . . . . . . . . . 10 (𝑐𝐶 ↦ (𝐹𝑐)) = (𝑐𝐶 ↦ (𝐹𝑐))
2120elrnmpt 5664 . . . . . . . . 9 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
226, 19, 213syl 18 . . . . . . . 8 (𝜑 → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2322ad3antrrr 717 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2418, 23mpbird 249 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)))
25 imaeq2 5760 . . . . . . . . 9 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
2625cbvmptv 5022 . . . . . . . 8 (𝑎𝐶 ↦ (𝐹𝑎)) = (𝑐𝐶 ↦ (𝐹𝑐))
2726rneqi 5643 . . . . . . 7 ran (𝑎𝐶 ↦ (𝐹𝑎)) = ran (𝑐𝐶 ↦ (𝐹𝑐))
2811, 27eqtri 2796 . . . . . 6 𝐷 = ran (𝑐𝐶 ↦ (𝐹𝑐))
2924, 28syl6eleqr 2871 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ 𝐷)
309ffnd 6339 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
3130ad3antrrr 717 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝐹 Fn 𝑋)
32 fbelss 22135 . . . . . . . . 9 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑎𝐶) → 𝑎𝑋)
334, 32sylan 572 . . . . . . . 8 ((𝜑𝑎𝐶) → 𝑎𝑋)
3433ad4ant13 738 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝑋)
35 fmucndlem 22593 . . . . . . 7 ((𝐹 Fn 𝑋𝑎𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
3631, 34, 35syl2anc 576 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
37 eqid 2772 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
3837mpofun 7086 . . . . . . . 8 Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
39 funimass2 6264 . . . . . . . 8 ((Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4038, 39mpan 677 . . . . . . 7 ((𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4140adantl 474 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4236, 41eqsstr3d 3892 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣)
43 id 22 . . . . . . . 8 (𝑏 = (𝐹𝑎) → 𝑏 = (𝐹𝑎))
4443sqxpeqd 5432 . . . . . . 7 (𝑏 = (𝐹𝑎) → (𝑏 × 𝑏) = ((𝐹𝑎) × (𝐹𝑎)))
4544sseq1d 3884 . . . . . 6 (𝑏 = (𝐹𝑎) → ((𝑏 × 𝑏) ⊆ 𝑣 ↔ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣))
4645rspcev 3529 . . . . 5 (((𝐹𝑎) ∈ 𝐷 ∧ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
4729, 42, 46syl2anc 576 . . . 4 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
481adantr 473 . . . . 5 ((𝜑𝑣𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
492adantr 473 . . . . 5 ((𝜑𝑣𝑉) → 𝐶 ∈ (CauFilu𝑈))
505adantr 473 . . . . . 6 ((𝜑𝑣𝑉) → 𝑉 ∈ (UnifOn‘𝑌))
516adantr 473 . . . . . 6 ((𝜑𝑣𝑉) → 𝐹 ∈ (𝑈 Cnu𝑉))
52 simpr 477 . . . . . 6 ((𝜑𝑣𝑉) → 𝑣𝑉)
53 nfcv 2926 . . . . . . 7 𝑠⟨(𝐹𝑥), (𝐹𝑦)⟩
54 nfcv 2926 . . . . . . 7 𝑡⟨(𝐹𝑥), (𝐹𝑦)⟩
55 nfcv 2926 . . . . . . 7 𝑥⟨(𝐹𝑠), (𝐹𝑡)⟩
56 nfcv 2926 . . . . . . 7 𝑦⟨(𝐹𝑠), (𝐹𝑡)⟩
57 simpl 475 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑥 = 𝑠)
5857fveq2d 6497 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑥) = (𝐹𝑠))
59 simpr 477 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑦 = 𝑡)
6059fveq2d 6497 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑦) = (𝐹𝑡))
6158, 60opeq12d 4679 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 𝑡) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6253, 54, 55, 56, 61cbvmpo 7058 . . . . . 6 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑠𝑋, 𝑡𝑋 ↦ ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6348, 50, 51, 52, 62ucnprima 22584 . . . . 5 ((𝜑𝑣𝑉) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈)
64 cfiluexsm 22592 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈) ∧ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6548, 49, 63, 64syl3anc 1351 . . . 4 ((𝜑𝑣𝑉) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6647, 65r19.29a 3228 . . 3 ((𝜑𝑣𝑉) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
6766ralrimiva 3126 . 2 (𝜑 → ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
68 iscfilu 22590 . . 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 700 1 (𝜑𝐷 ∈ (CauFilu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wral 3082  wrex 3083  Vcvv 3409  wss 3825  cop 4441   class class class wbr 4923  cmpt 5002   × cxp 5398  ccnv 5399  ran crn 5401  cima 5403  Fun wfun 6176   Fn wfn 6177  wf 6178  cfv 6182  (class class class)co 6970  cmpo 6972  fBascfbas 20225  UnifOncust 22501   Cnucucn 22577  CauFiluccfilu 22588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-map 8200  df-fbas 20234  df-ust 22502  df-ucn 22578  df-cfilu 22589
This theorem is referenced by:  ucnextcn  22606
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