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Theorem fmco 23985
Description: Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
fmco (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑋 FilMap (𝐹𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)))

Proof of Theorem fmco
Dummy variables 𝑡 𝑠 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1192 . . . . . . . . . . 11 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2 ssfg 23896 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑍) → 𝐵 ⊆ (𝑍filGen𝐵))
31, 2syl 17 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ⊆ (𝑍filGen𝐵))
43sseld 3994 . . . . . . . . 9 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑢𝐵𝑢 ∈ (𝑍filGen𝐵)))
5 simpl2 1191 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑌𝑊)
6 simprr 773 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐺:𝑍𝑌)
7 eqid 2735 . . . . . . . . . . . 12 (𝑍filGen𝐵) = (𝑍filGen𝐵)
87imaelfm 23975 . . . . . . . . . . 11 (((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
98ex 412 . . . . . . . . . 10 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
105, 1, 6, 9syl3anc 1370 . . . . . . . . 9 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
114, 10syld 47 . . . . . . . 8 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑢𝐵 → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
1211imp 406 . . . . . . 7 ((((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) ∧ 𝑢𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13 imaeq2 6076 . . . . . . . . . . 11 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = (𝐹 “ (𝐺𝑢)))
14 imaco 6273 . . . . . . . . . . 11 ((𝐹𝐺) “ 𝑢) = (𝐹 “ (𝐺𝑢))
1513, 14eqtr4di 2793 . . . . . . . . . 10 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = ((𝐹𝐺) “ 𝑢))
1615sseq1d 4027 . . . . . . . . 9 (𝑡 = (𝐺𝑢) → ((𝐹𝑡) ⊆ 𝑠 ↔ ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
1716rspcev 3622 . . . . . . . 8 (((𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)
1817ex 412 . . . . . . 7 ((𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) → (((𝐹𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
1912, 18syl 17 . . . . . 6 ((((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) ∧ 𝑢𝐵) → (((𝐹𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
2019rexlimdva 3153 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
21 elfm 23971 . . . . . . . 8 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
225, 1, 6, 21syl3anc 1370 . . . . . . 7 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
23 sstr2 4002 . . . . . . . . . . 11 (((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
24 imass2 6123 . . . . . . . . . . . 12 ((𝐺𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺𝑢)) ⊆ (𝐹𝑡))
2514, 24eqsstrid 4044 . . . . . . . . . . 11 ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡))
2623, 25syl11 33 . . . . . . . . . 10 ((𝐹𝑡) ⊆ 𝑠 → ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2726reximdv 3168 . . . . . . . . 9 ((𝐹𝑡) ⊆ 𝑠 → (∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2827com12 32 . . . . . . . 8 (∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡 → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2928adantl 481 . . . . . . 7 ((𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
3022, 29biimtrdi 253 . . . . . 6 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3130rexlimdv 3151 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
3220, 31impbid 212 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 ↔ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
3332anbi2d 630 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠) ↔ (𝑠𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)))
34 simpl1 1190 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑋𝑉)
35 fco 6761 . . . . 5 ((𝐹:𝑌𝑋𝐺:𝑍𝑌) → (𝐹𝐺):𝑍𝑋)
3635adantl 481 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝐹𝐺):𝑍𝑋)
37 elfm 23971 . . . 4 ((𝑋𝑉𝐵 ∈ (fBas‘𝑍) ∧ (𝐹𝐺):𝑍𝑋) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3834, 1, 36, 37syl3anc 1370 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
39 fmfil 23968 . . . . . 6 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
405, 1, 6, 39syl3anc 1370 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41 filfbas 23872 . . . . 5 (((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
4240, 41syl 17 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
43 simprl 771 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐹:𝑌𝑋)
44 elfm 23971 . . . 4 ((𝑋𝑉 ∧ ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)) ↔ (𝑠𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)))
4534, 42, 43, 44syl3anc 1370 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)) ↔ (𝑠𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)))
4633, 38, 453bitr4d 311 . 2 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ 𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵))))
4746eqrdv 2733 1 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑋 FilMap (𝐹𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  wss 3963  cima 5692  ccom 5693  wf 6559  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:  ufldom  23986  flfcnp  24028
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