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Theorem fmfnfmlem2 22038
Description: Lemma for fmfnfm 22041. (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 717 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝐿 ∈ (Fil‘𝑋))
3 simplr 785 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑥𝐿)
4 fmfnfm.fm . . . . . . . 8 (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
5 fmfnfm.f . . . . . . . . . 10 (𝜑𝐹:𝑌𝑋)
6 ffn 6223 . . . . . . . . . . 11 (𝐹:𝑌𝑋𝐹 Fn 𝑌)
7 dffn4 6304 . . . . . . . . . . 11 (𝐹 Fn 𝑌𝐹:𝑌onto→ran 𝐹)
86, 7sylib 209 . . . . . . . . . 10 (𝐹:𝑌𝑋𝐹:𝑌onto→ran 𝐹)
9 foima 6303 . . . . . . . . . 10 (𝐹:𝑌onto→ran 𝐹 → (𝐹𝑌) = ran 𝐹)
105, 8, 93syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝑌) = ran 𝐹)
11 filtop 21938 . . . . . . . . . . 11 (𝐿 ∈ (Fil‘𝑋) → 𝑋𝐿)
121, 11syl 17 . . . . . . . . . 10 (𝜑𝑋𝐿)
13 fmfnfm.b . . . . . . . . . 10 (𝜑𝐵 ∈ (fBas‘𝑌))
14 fgcl 21961 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
15 filtop 21938 . . . . . . . . . . 11 ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
1613, 14, 153syl 18 . . . . . . . . . 10 (𝜑𝑌 ∈ (𝑌filGen𝐵))
17 eqid 2765 . . . . . . . . . . 11 (𝑌filGen𝐵) = (𝑌filGen𝐵)
1817imaelfm 22034 . . . . . . . . . 10 (((𝑋𝐿𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
1912, 13, 5, 16, 18syl31anc 1492 . . . . . . . . 9 (𝜑 → (𝐹𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
2010, 19eqeltrrd 2845 . . . . . . . 8 (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
214, 20sseldd 3762 . . . . . . 7 (𝜑 → ran 𝐹𝐿)
2221ad2antrr 717 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ran 𝐹𝐿)
23 filin 21937 . . . . . 6 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥𝐿 ∧ ran 𝐹𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
242, 3, 22, 23syl3anc 1490 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
25 simprr 789 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝑋)
26 elin 3958 . . . . . . 7 (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦𝑥𝑦 ∈ ran 𝐹))
27 fvelrnb 6432 . . . . . . . . . . . . 13 (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
285, 6, 273syl 18 . . . . . . . . . . . 12 (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
2928ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧𝑌 (𝐹𝑧) = 𝑦))
305ffund 6227 . . . . . . . . . . . . . . . . 17 (𝜑 → Fun 𝐹)
3130ad2antrr 717 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → Fun 𝐹)
32 simprr 789 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧𝑌)
335fdmd 6232 . . . . . . . . . . . . . . . . . 18 (𝜑 → dom 𝐹 = 𝑌)
3433ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → dom 𝐹 = 𝑌)
3532, 34eleqtrrd 2847 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → 𝑧 ∈ dom 𝐹)
36 fvimacnv 6522 . . . . . . . . . . . . . . . 16 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
3731, 35, 36syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥𝑧 ∈ (𝐹𝑥)))
38 cnvimass 5667 . . . . . . . . . . . . . . . . 17 (𝐹𝑥) ⊆ dom 𝐹
39 funfvima2 6686 . . . . . . . . . . . . . . . . 17 ((Fun 𝐹 ∧ (𝐹𝑥) ⊆ dom 𝐹) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥))))
4031, 38, 39sylancl 580 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥))))
41 ssel 3755 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → ((𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥)) → (𝐹𝑧) ∈ 𝑡))
4241ad2antrl 719 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ (𝐹 “ (𝐹𝑥)) → (𝐹𝑧) ∈ 𝑡))
4340, 42syld 47 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → (𝑧 ∈ (𝐹𝑥) → (𝐹𝑧) ∈ 𝑡))
4437, 43sylbid 231 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) ∈ 𝑥 → (𝐹𝑧) ∈ 𝑡))
45 eleq1 2832 . . . . . . . . . . . . . . 15 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑥𝑦𝑥))
46 eleq1 2832 . . . . . . . . . . . . . . 15 ((𝐹𝑧) = 𝑦 → ((𝐹𝑧) ∈ 𝑡𝑦𝑡))
4745, 46imbi12d 335 . . . . . . . . . . . . . 14 ((𝐹𝑧) = 𝑦 → (((𝐹𝑧) ∈ 𝑥 → (𝐹𝑧) ∈ 𝑡) ↔ (𝑦𝑥𝑦𝑡)))
4844, 47syl5ibcom 236 . . . . . . . . . . . . 13 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑧𝑌)) → ((𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡)))
4948expr 448 . . . . . . . . . . . 12 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑧𝑌 → ((𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡))))
5049rexlimdv 3177 . . . . . . . . . . 11 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (∃𝑧𝑌 (𝐹𝑧) = 𝑦 → (𝑦𝑥𝑦𝑡)))
5129, 50sylbid 231 . . . . . . . . . 10 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 → (𝑦𝑥𝑦𝑡)))
5251com23 86 . . . . . . . . 9 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → (𝑦𝑥 → (𝑦 ∈ ran 𝐹𝑦𝑡)))
5352impd 398 . . . . . . . 8 (((𝜑𝑥𝐿) ∧ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡) → ((𝑦𝑥𝑦 ∈ ran 𝐹) → 𝑦𝑡))
5453adantrr 708 . . . . . . 7 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → ((𝑦𝑥𝑦 ∈ ran 𝐹) → 𝑦𝑡))
5526, 54syl5bi 233 . . . . . 6 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑦 ∈ (𝑥 ∩ ran 𝐹) → 𝑦𝑡))
5655ssrdv 3767 . . . . 5 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ 𝑡)
57 filss 21936 . . . . 5 ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝑥 ∩ ran 𝐹) ∈ 𝐿𝑡𝑋 ∧ (𝑥 ∩ ran 𝐹) ⊆ 𝑡)) → 𝑡𝐿)
582, 24, 25, 56, 57syl13anc 1491 . . . 4 (((𝜑𝑥𝐿) ∧ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡𝑡𝑋)) → 𝑡𝐿)
5958exp32 411 . . 3 ((𝜑𝑥𝐿) → ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)))
60 imaeq2 5644 . . . . 5 (𝑠 = (𝐹𝑥) → (𝐹𝑠) = (𝐹 “ (𝐹𝑥)))
6160sseq1d 3792 . . . 4 (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝐹𝑥)) ⊆ 𝑡))
6261imbi1d 332 . . 3 (𝑠 = (𝐹𝑥) → (((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿)) ↔ ((𝐹 “ (𝐹𝑥)) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
6359, 62syl5ibrcom 238 . 2 ((𝜑𝑥𝐿) → (𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
6463rexlimdva 3178 1 (𝜑 → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  cin 3731  wss 3732  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  Fun wfun 6062   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  (class class class)co 6842  fBascfbas 20007  filGencfg 20008  Filcfil 21928   FilMap cfm 22016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-fbas 20016  df-fg 20017  df-fil 21929  df-fm 22021
This theorem is referenced by:  fmfnfmlem4  22040
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