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Theorem fmfnfmlem2 23964
Description: Lemma for fmfnfm 23967. (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 726 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝐿 ∈ (Fil‘𝑋))
3 simplr 768 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑥𝐿)
4 fmfnfm.fm . . . . . . . 8 (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
5 fmfnfm.f . . . . . . . . . 10 (𝜑𝐹:𝑌𝑋)
6 ffn 6735 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
7 dffn4 6825 . . . . . . . . . . 11 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
86, 7sylib 218 . . . . . . . . . 10 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
9 foima 6824 . . . . . . . . . 10 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
105, 8, 93syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝑌) = ran 𝐹)
11 filtop 23864 . . . . . . . . . . 11 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
121, 11syl 17 . . . . . . . . . 10 (𝜑𝑋𝐿)
13 fmfnfm.b . . . . . . . . . 10 (𝜑𝐵 ∈ (fBas‘𝑌))
14 fgcl 23887 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
15 filtop 23864 . . . . . . . . . . 11 ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
1613, 14, 153syl 18 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝑌filGen𝐵))
17 eqid 2736 . . . . . . . . . . 11 (𝑌filGen𝐵) = (𝑌filGen𝐵)
1817imaelfm 23960 . . . . . . . . . 10 (((𝑋𝐿𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
1912, 13, 5, 16, 18syl31anc 1374 . . . . . . . . 9 (𝜑 → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
2010, 19eqeltrrd 2841 . . . . . . . 8 (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
214, 20sseldd 3983 . . . . . . 7 (𝜑 → ran 𝐹𝐿)
2221ad2antrr 726 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ran 𝐹𝐿)
23 filin 23863 . . . . . 6 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
242, 3, 22, 23syl3anc 1372 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
25 simprr 772 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝑋)
26 elin 3966 . . . . . . 7 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
27 fvelrnb 6968 . . . . . . . . . . . 12 (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
285, 6, 273syl 18 . . . . . . . . . . 11 (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
2928ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
305ffund 6739 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
3130ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → Fun 𝐹)
32 simprr 772 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧𝑌)
335fdmd 6745 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐹 = 𝑌)
3433ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → dom 𝐹 = 𝑌)
3532, 34eleqtrrd 2843 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧 ∈ dom 𝐹)
36 fvimacnv 7072 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
3731, 35, 36syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
38 cnvimass 6099 . . . . . . . . . . . . . . . 16 (𝐹𝑥) ⊆ dom 𝐹
39 funfvima2 7252 . . . . . . . . . . . . . . . 16 ((Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥))))
4031, 38, 39sylancl 586 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥))))
41 ssel 3976 . . . . . . . . . . . . . . . 16 ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → ((𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥)) → (𝐹𝑧) ∈ 𝑡))
4241ad2antrl 728 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥)) → (𝐹𝑧) ∈ 𝑡))
4340, 42syld 47 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ 𝑡))
4437, 43sylbid 240 . . . . . . . . . . . . 13 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥 → (𝐹𝑧) ∈ 𝑡))
45 eleq1 2828 . . . . . . . . . . . . . 14 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
46 eleq1 2828 . . . . . . . . . . . . . 14 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑡𝑦𝑡))
4745, 46imbi12d 344 . . . . . . . . . . . . 13 ((𝐹𝑧) = 𝑦 → (((𝐹𝑧) ∈ 𝑥 → (𝐹𝑧) ∈ 𝑡) ↔ (𝑦𝑥𝑦𝑡)))
4844, 47syl5ibcom 245 . . . . . . . . . . . 12 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡)))
4948expr 456 . . . . . . . . . . 11 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑧𝑌 → ((𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡))))
5049rexlimdv 3152 . . . . . . . . . 10 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (∃𝑧𝑌 (𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡)))
5129, 50sylbid 240 . . . . . . . . 9 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 → (𝑦𝑥𝑦𝑡)))
5251impcomd 411 . . . . . . . 8 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → ((𝑦𝑥𝑦 ∈ ran 𝐹) → 𝑦𝑡))
5352adantrr 717 . . . . . . 7 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝑦𝑥𝑦 ∈ ran 𝐹) → 𝑦𝑡))
5426, 53biimtrid 242 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑦 ∈ (𝑥 ∩ ran 𝐹) → 𝑦𝑡))
5554ssrdv 3988 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ 𝑡)
56 filss 23862 . . . . 5 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝑥 ∩ ran 𝐹) ∈ 𝐿𝑡𝑋 ∧ (𝑥 ∩ ran 𝐹) ⊆ 𝑡)) → 𝑡𝐿)
572, 24, 25, 55, 56syl13anc 1373 . . . 4 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝐿)
5857exp32 420 . . 3 ((𝜑𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
59 imaeq2 6073 . . . . 5 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
6059sseq1d 4014 . . . 4 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
6160imbi1d 341 . . 3 (𝑠 = (𝐹𝑥) → (((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)) ↔ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
6258, 61syl5ibrcom 247 . 2 ((𝜑𝑥𝐿) → (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
6362rexlimdva 3154 1 (𝜑 → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wrex 3069  cin 3949  wss 3950  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687  Fun wfun 6554   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  fBascfbas 21353  filGencfg 21354  Filcfil 23854   FilMap cfm 23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  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 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-fg 21363  df-fil 23855  df-fm 23947
This theorem is referenced by:  fmfnfmlem4  23966
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