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Theorem fmfnfmlem2 21806
Description: Lemma for fmfnfm 21809. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypotheses
Ref Expression
fmfnfm.b (𝜑𝐵 ∈ (fBas‘𝑌))
fmfnfm.l (𝜑𝐿 ∈ (Fil‘𝑋))
fmfnfm.f (𝜑𝐹:𝑌𝑋)
fmfnfm.fm (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
Assertion
Ref Expression
fmfnfmlem2 (𝜑 → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
Distinct variable groups:   𝑡,𝑠,𝑥,𝐵   𝐹,𝑠,𝑡,𝑥   𝐿,𝑠,𝑡,𝑥   𝜑,𝑠,𝑡,𝑥   𝑋,𝑠,𝑡,𝑥   𝑌,𝑠,𝑡,𝑥

Proof of Theorem fmfnfmlem2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmfnfm.l . . . . . 6 (𝜑𝐿 ∈ (Fil‘𝑋))
21ad2antrr 762 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝐿 ∈ (Fil‘𝑋))
3 simplr 807 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑥𝐿)
4 fmfnfm.fm . . . . . . . 8 (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
5 fmfnfm.f . . . . . . . . . 10 (𝜑𝐹:𝑌𝑋)
6 ffn 6083 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
7 dffn4 6159 . . . . . . . . . . 11 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
86, 7sylib 208 . . . . . . . . . 10 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
9 foima 6158 . . . . . . . . . 10 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
105, 8, 93syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝑌) = ran 𝐹)
11 filtop 21706 . . . . . . . . . . 11 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
121, 11syl 17 . . . . . . . . . 10 (𝜑𝑋𝐿)
13 fmfnfm.b . . . . . . . . . 10 (𝜑𝐵 ∈ (fBas‘𝑌))
14 fgcl 21729 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
15 filtop 21706 . . . . . . . . . . 11 ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
1613, 14, 153syl 18 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝑌filGen𝐵))
17 eqid 2651 . . . . . . . . . . 11 (𝑌filGen𝐵) = (𝑌filGen𝐵)
1817imaelfm 21802 . . . . . . . . . 10 (((𝑋𝐿𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
1912, 13, 5, 16, 18syl31anc 1369 . . . . . . . . 9 (𝜑 → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
2010, 19eqeltrrd 2731 . . . . . . . 8 (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
214, 20sseldd 3637 . . . . . . 7 (𝜑 → ran 𝐹𝐿)
2221ad2antrr 762 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ran 𝐹𝐿)
23 filin 21705 . . . . . 6 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
242, 3, 22, 23syl3anc 1366 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
25 simprr 811 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝑋)
26 elin 3829 . . . . . . 7 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
27 fvelrnb 6282 . . . . . . . . . . . . 13 (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
285, 6, 273syl 18 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
2928ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
30 ffun 6086 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑌𝑋 → Fun 𝐹)
315, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝐹)
3231ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → Fun 𝐹)
33 simprr 811 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧𝑌)
34 fdm 6089 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝑌𝑋 → dom 𝐹 = 𝑌)
355, 34syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = 𝑌)
3635ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → dom 𝐹 = 𝑌)
3733, 36eleqtrrd 2733 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧 ∈ dom 𝐹)
38 fvimacnv 6372 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
3932, 37, 38syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
40 cnvimass 5520 . . . . . . . . . . . . . . . . 17 (𝐹𝑥) ⊆ dom 𝐹
41 funfvima2 6533 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥))))
4232, 40, 41sylancl 695 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥))))
43 ssel 3630 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → ((𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥)) → (𝐹𝑧) ∈ 𝑡))
4443ad2antrl 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥)) → (𝐹𝑧) ∈ 𝑡))
4542, 44syld 47 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ 𝑡))
4639, 45sylbid 230 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥 → (𝐹𝑧) ∈ 𝑡))
47 eleq1 2718 . . . . . . . . . . . . . . 15 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
48 eleq1 2718 . . . . . . . . . . . . . . 15 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑡𝑦𝑡))
4947, 48imbi12d 333 . . . . . . . . . . . . . 14 ((𝐹𝑧) = 𝑦 → (((𝐹𝑧) ∈ 𝑥 → (𝐹𝑧) ∈ 𝑡) ↔ (𝑦𝑥𝑦𝑡)))
5046, 49syl5ibcom 235 . . . . . . . . . . . . 13 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡)))
5150expr 642 . . . . . . . . . . . 12 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑧𝑌 → ((𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡))))
5251rexlimdv 3059 . . . . . . . . . . 11 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (∃𝑧𝑌 (𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡)))
5329, 52sylbid 230 . . . . . . . . . 10 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 → (𝑦𝑥𝑦𝑡)))
5453com23 86 . . . . . . . . 9 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦𝑥 → (𝑦 ∈ ran 𝐹𝑦𝑡)))
5554impd 446 . . . . . . . 8 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → ((𝑦𝑥𝑦 ∈ ran 𝐹) → 𝑦𝑡))
5655adantrr 753 . . . . . . 7 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝑦𝑥𝑦 ∈ ran 𝐹) → 𝑦𝑡))
5726, 56syl5bi 232 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑦 ∈ (𝑥 ∩ ran 𝐹) → 𝑦𝑡))
5857ssrdv 3642 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ 𝑡)
59 filss 21704 . . . . 5 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝑥 ∩ ran 𝐹) ∈ 𝐿𝑡𝑋 ∧ (𝑥 ∩ ran 𝐹) ⊆ 𝑡)) → 𝑡𝐿)
602, 24, 25, 58, 59syl13anc 1368 . . . 4 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝐿)
6160exp32 630 . . 3 ((𝜑𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
62 imaeq2 5497 . . . . 5 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
6362sseq1d 3665 . . . 4 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
6463imbi1d 330 . . 3 (𝑠 = (𝐹𝑥) → (((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)) ↔ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
6561, 64syl5ibrcom 237 . 2 ((𝜑𝑥𝐿) → (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
6665rexlimdva 3060 1 (𝜑 → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cin 3606  wss 3607  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  fBascfbas 19782  filGencfg 19783  Filcfil 21696   FilMap cfm 21784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fbas 19791  df-fg 19792  df-fil 21697  df-fm 21789
This theorem is referenced by:  fmfnfmlem4  21808
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