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Theorem fmucnd 22077
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 22074 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈)) → 𝐶 ∈ (fBas‘𝑋))
41, 2, 3syl2anc 692 . . 3 (𝜑𝐶 ∈ (fBas‘𝑋))
5 fmucnd.2 . . . 4 (𝜑𝑉 ∈ (UnifOn‘𝑌))
6 fmucnd.3 . . . 4 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
7 isucn 22063 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
87simprbda 652 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) ∧ 𝐹 ∈ (𝑈 Cnu𝑉)) → 𝐹:𝑋𝑌)
91, 5, 6, 8syl21anc 1323 . . 3 (𝜑𝐹:𝑋𝑌)
105elfvexd 6209 . . 3 (𝜑𝑌 ∈ V)
11 fmucnd.5 . . . 4 𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))
1211fbasrn 21669 . . 3 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌 ∈ V) → 𝐷 ∈ (fBas‘𝑌))
134, 9, 10, 12syl3anc 1324 . 2 (𝜑𝐷 ∈ (fBas‘𝑌))
14 simplr 791 . . . . . . . 8 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝐶)
15 eqid 2620 . . . . . . . 8 (𝐹𝑎) = (𝐹𝑎)
16 imaeq2 5450 . . . . . . . . . 10 (𝑐 = 𝑎 → (𝐹𝑐) = (𝐹𝑎))
1716eqeq2d 2630 . . . . . . . . 9 (𝑐 = 𝑎 → ((𝐹𝑎) = (𝐹𝑐) ↔ (𝐹𝑎) = (𝐹𝑎)))
1817rspcev 3304 . . . . . . . 8 ((𝑎𝐶 ∧ (𝐹𝑎) = (𝐹𝑎)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
1914, 15, 18sylancl 693 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐))
20 imaexg 7088 . . . . . . . . 9 (𝐹 ∈ (𝑈 Cnu𝑉) → (𝐹𝑎) ∈ V)
21 eqid 2620 . . . . . . . . . 10 (𝑐𝐶 ↦ (𝐹𝑐)) = (𝑐𝐶 ↦ (𝐹𝑐))
2221elrnmpt 5361 . . . . . . . . 9 ((𝐹𝑎) ∈ V → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
236, 20, 223syl 18 . . . . . . . 8 (𝜑 → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2423ad3antrrr 765 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)) ↔ ∃𝑐𝐶 (𝐹𝑎) = (𝐹𝑐)))
2519, 24mpbird 247 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ ran (𝑐𝐶 ↦ (𝐹𝑐)))
26 imaeq2 5450 . . . . . . . . 9 (𝑎 = 𝑐 → (𝐹𝑎) = (𝐹𝑐))
2726cbvmptv 4741 . . . . . . . 8 (𝑎𝐶 ↦ (𝐹𝑎)) = (𝑐𝐶 ↦ (𝐹𝑐))
2827rneqi 5341 . . . . . . 7 ran (𝑎𝐶 ↦ (𝐹𝑎)) = ran (𝑐𝐶 ↦ (𝐹𝑐))
2911, 28eqtri 2642 . . . . . 6 𝐷 = ran (𝑐𝐶 ↦ (𝐹𝑐))
3025, 29syl6eleqr 2710 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → (𝐹𝑎) ∈ 𝐷)
31 ffn 6032 . . . . . . . . 9 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
329, 31syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
3332ad3antrrr 765 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝐹 Fn 𝑋)
34 simplll 797 . . . . . . . 8 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝜑)
35 fbelss 21618 . . . . . . . . 9 ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑎𝐶) → 𝑎𝑋)
364, 35sylan 488 . . . . . . . 8 ((𝜑𝑎𝐶) → 𝑎𝑋)
3734, 14, 36syl2anc 692 . . . . . . 7 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → 𝑎𝑋)
38 fmucndlem 22076 . . . . . . 7 ((𝐹 Fn 𝑋𝑎𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
3933, 37, 38syl2anc 692 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹𝑎) × (𝐹𝑎)))
40 eqid 2620 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
4140mpt2fun 6747 . . . . . . . 8 Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
42 funimass2 5960 . . . . . . . 8 ((Fun (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4341, 42mpan 705 . . . . . . 7 ((𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4443adantl 482 . . . . . 6 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
4539, 44eqsstr3d 3632 . . . . 5 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣)
46 id 22 . . . . . . . 8 (𝑏 = (𝐹𝑎) → 𝑏 = (𝐹𝑎))
4746sqxpeqd 5131 . . . . . . 7 (𝑏 = (𝐹𝑎) → (𝑏 × 𝑏) = ((𝐹𝑎) × (𝐹𝑎)))
4847sseq1d 3624 . . . . . 6 (𝑏 = (𝐹𝑎) → ((𝑏 × 𝑏) ⊆ 𝑣 ↔ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣))
4948rspcev 3304 . . . . 5 (((𝐹𝑎) ∈ 𝐷 ∧ ((𝐹𝑎) × (𝐹𝑎)) ⊆ 𝑣) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
5030, 45, 49syl2anc 692 . . . 4 ((((𝜑𝑣𝑉) ∧ 𝑎𝐶) ∧ (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣)) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
511adantr 481 . . . . 5 ((𝜑𝑣𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
522adantr 481 . . . . 5 ((𝜑𝑣𝑉) → 𝐶 ∈ (CauFilu𝑈))
535adantr 481 . . . . . 6 ((𝜑𝑣𝑉) → 𝑉 ∈ (UnifOn‘𝑌))
546adantr 481 . . . . . 6 ((𝜑𝑣𝑉) → 𝐹 ∈ (𝑈 Cnu𝑉))
55 simpr 477 . . . . . 6 ((𝜑𝑣𝑉) → 𝑣𝑉)
56 nfcv 2762 . . . . . . 7 𝑠⟨(𝐹𝑥), (𝐹𝑦)⟩
57 nfcv 2762 . . . . . . 7 𝑡⟨(𝐹𝑥), (𝐹𝑦)⟩
58 nfcv 2762 . . . . . . 7 𝑥⟨(𝐹𝑠), (𝐹𝑡)⟩
59 nfcv 2762 . . . . . . 7 𝑦⟨(𝐹𝑠), (𝐹𝑡)⟩
60 simpl 473 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑥 = 𝑠)
6160fveq2d 6182 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑥) = (𝐹𝑠))
62 simpr 477 . . . . . . . . 9 ((𝑥 = 𝑠𝑦 = 𝑡) → 𝑦 = 𝑡)
6362fveq2d 6182 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 𝑡) → (𝐹𝑦) = (𝐹𝑡))
6461, 63opeq12d 4401 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 𝑡) → ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6556, 57, 58, 59, 64cbvmpt2 6719 . . . . . 6 (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑠𝑋, 𝑡𝑋 ↦ ⟨(𝐹𝑠), (𝐹𝑡)⟩)
6651, 53, 54, 55, 65ucnprima 22067 . . . . 5 ((𝜑𝑣𝑉) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈)
67 cfiluexsm 22075 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu𝑈) ∧ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣) ∈ 𝑈) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6851, 52, 66, 67syl3anc 1324 . . . 4 ((𝜑𝑣𝑉) → ∃𝑎𝐶 (𝑎 × 𝑎) ⊆ ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ 𝑣))
6950, 68r19.29a 3074 . . 3 ((𝜑𝑣𝑉) → ∃𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
7069ralrimiva 2963 . 2 (𝜑 → ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
71 iscfilu 22073 . . 3 (𝑉 ∈ (UnifOn‘𝑌) → (𝐷 ∈ (CauFilu𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
725, 71syl 17 . 2 (𝜑 → (𝐷 ∈ (CauFilu𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣𝑉𝑏𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
7313, 70, 72mpbir2and 956 1 (𝜑𝐷 ∈ (CauFilu𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  wss 3567  cop 4174   class class class wbr 4644  cmpt 4720   × cxp 5102  ccnv 5103  ran crn 5105  cima 5107  Fun wfun 5870   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  fBascfbas 19715  UnifOncust 21984   Cnucucn 22060  CauFiluccfilu 22071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-fbas 19724  df-ust 21985  df-ucn 22061  df-cfilu 22072
This theorem is referenced by:  ucnextcn  22089
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