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Theorem fmfnfmlem2 23979
Description: Lemma for fmfnfm 23982. (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 769 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑥𝐿)
4 fmfnfm.fm . . . . . . . 8 (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
5 fmfnfm.f . . . . . . . . . 10 (𝜑𝐹:𝑌𝑋)
6 ffn 6737 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
7 dffn4 6827 . . . . . . . . . . 11 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
86, 7sylib 218 . . . . . . . . . 10 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
9 foima 6826 . . . . . . . . . 10 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
105, 8, 93syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝑌) = ran 𝐹)
11 filtop 23879 . . . . . . . . . . 11 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
121, 11syl 17 . . . . . . . . . 10 (𝜑𝑋𝐿)
13 fmfnfm.b . . . . . . . . . 10 (𝜑𝐵 ∈ (fBas‘𝑌))
14 fgcl 23902 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
15 filtop 23879 . . . . . . . . . . 11 ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
1613, 14, 153syl 18 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝑌filGen𝐵))
17 eqid 2735 . . . . . . . . . . 11 (𝑌filGen𝐵) = (𝑌filGen𝐵)
1817imaelfm 23975 . . . . . . . . . 10 (((𝑋𝐿𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
1912, 13, 5, 16, 18syl31anc 1372 . . . . . . . . 9 (𝜑 → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
2010, 19eqeltrrd 2840 . . . . . . . 8 (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
214, 20sseldd 3996 . . . . . . 7 (𝜑 → ran 𝐹𝐿)
2221ad2antrr 726 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ran 𝐹𝐿)
23 filin 23878 . . . . . 6 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
242, 3, 22, 23syl3anc 1370 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
25 simprr 773 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝑋)
26 elin 3979 . . . . . . 7 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
27 fvelrnb 6969 . . . . . . . . . . . 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 6741 . . . . . . . . . . . . . . . 16 (𝜑 → Fun 𝐹)
3130ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → Fun 𝐹)
32 simprr 773 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧𝑌)
335fdmd 6747 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐹 = 𝑌)
3433ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → dom 𝐹 = 𝑌)
3532, 34eleqtrrd 2842 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧 ∈ dom 𝐹)
36 fvimacnv 7073 . . . . . . . . . . . . . . 15 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
3731, 35, 36syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
38 cnvimass 6102 . . . . . . . . . . . . . . . 16 (𝐹𝑥) ⊆ dom 𝐹
39 funfvima2 7251 . . . . . . . . . . . . . . . 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 3989 . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . 14 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
46 eleq1 2827 . . . . . . . . . . . . . 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 3151 . . . . . . . . . 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 4001 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ 𝑡)
56 filss 23877 . . . . 5 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝑥 ∩ ran 𝐹) ∈ 𝐿𝑡𝑋 ∧ (𝑥 ∩ ran 𝐹) ⊆ 𝑡)) → 𝑡𝐿)
572, 24, 25, 55, 56syl13anc 1371 . . . 4 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝐿)
5857exp32 420 . . 3 ((𝜑𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
59 imaeq2 6076 . . . . 5 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
6059sseq1d 4027 . . . 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 3153 1 (𝜑 → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cin 3962  wss 3963  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  Filcfil 23869   FilMap cfm 23957
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
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-reu 3379  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-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fil 23870  df-fm 23962
This theorem is referenced by:  fmfnfmlem4  23981
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