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Theorem fmco 23969
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 1194 . . . . . . . . . . 11 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2 ssfg 23880 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑍) → 𝐵 ⊆ (𝑍filGen𝐵))
31, 2syl 17 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ⊆ (𝑍filGen𝐵))
43sseld 3982 . . . . . . . . 9 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑢𝐵𝑢 ∈ (𝑍filGen𝐵)))
5 simpl2 1193 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑌𝑊)
6 simprr 773 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐺:𝑍𝑌)
7 eqid 2737 . . . . . . . . . . . 12 (𝑍filGen𝐵) = (𝑍filGen𝐵)
87imaelfm 23959 . . . . . . . . . . 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 1373 . . . . . . . . 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 6074 . . . . . . . . . . 11 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = (𝐹 “ (𝐺𝑢)))
14 imaco 6271 . . . . . . . . . . 11 ((𝐹𝐺) “ 𝑢) = (𝐹 “ (𝐺𝑢))
1513, 14eqtr4di 2795 . . . . . . . . . 10 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = ((𝐹𝐺) “ 𝑢))
1615sseq1d 4015 . . . . . . . . 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 3155 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
21 elfm 23955 . . . . . . . 8 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
225, 1, 6, 21syl3anc 1373 . . . . . . 7 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
23 sstr2 3990 . . . . . . . . . . 11 (((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
24 imass2 6120 . . . . . . . . . . . 12 ((𝐺𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺𝑢)) ⊆ (𝐹𝑡))
2514, 24eqsstrid 4022 . . . . . . . . . . 11 ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡))
2623, 25syl11 33 . . . . . . . . . 10 ((𝐹𝑡) ⊆ 𝑠 → ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2726reximdv 3170 . . . . . . . . 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 3153 . . . . 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 1192 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑋𝑉)
35 fco 6760 . . . . 5 ((𝐹:𝑌𝑋𝐺:𝑍𝑌) → (𝐹𝐺):𝑍𝑋)
3635adantl 481 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝐹𝐺):𝑍𝑋)
37 elfm 23955 . . . 4 ((𝑋𝑉𝐵 ∈ (fBas‘𝑍) ∧ (𝐹𝐺):𝑍𝑋) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3834, 1, 36, 37syl3anc 1373 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
39 fmfil 23952 . . . . . 6 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
405, 1, 6, 39syl3anc 1373 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41 filfbas 23856 . . . . 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 23955 . . . 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 1373 . . 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 2735 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 1087   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cima 5688  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353  Filcfil 23853   FilMap cfm 23941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fg 21362  df-fil 23854  df-fm 23946
This theorem is referenced by:  ufldom  23970  flfcnp  24012
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