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Theorem fmco 21670
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 1064 . . . . . . . . . . 11 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2 ssfg 21581 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑍) → 𝐵 ⊆ (𝑍filGen𝐵))
31, 2syl 17 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ⊆ (𝑍filGen𝐵))
43sseld 3587 . . . . . . . . 9 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑢𝐵𝑢 ∈ (𝑍filGen𝐵)))
5 simpl2 1063 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑌𝑊)
6 simprr 795 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐺:𝑍𝑌)
7 eqid 2626 . . . . . . . . . . . 12 (𝑍filGen𝐵) = (𝑍filGen𝐵)
87imaelfm 21660 . . . . . . . . . . 11 (((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
98ex 450 . . . . . . . . . 10 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
105, 1, 6, 9syl3anc 1323 . . . . . . . . 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 445 . . . . . . 7 ((((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) ∧ 𝑢𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13 imaeq2 5425 . . . . . . . . . . 11 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = (𝐹 “ (𝐺𝑢)))
14 imaco 5602 . . . . . . . . . . 11 ((𝐹𝐺) “ 𝑢) = (𝐹 “ (𝐺𝑢))
1513, 14syl6eqr 2678 . . . . . . . . . 10 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = ((𝐹𝐺) “ 𝑢))
1615sseq1d 3616 . . . . . . . . 9 (𝑡 = (𝐺𝑢) → ((𝐹𝑡) ⊆ 𝑠 ↔ ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
1716rspcev 3300 . . . . . . . 8 (((𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)
1817ex 450 . . . . . . 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 3029 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
21 elfm 21656 . . . . . . . 8 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
225, 1, 6, 21syl3anc 1323 . . . . . . 7 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
23 sstr2 3595 . . . . . . . . . . 11 (((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
24 imass2 5464 . . . . . . . . . . . 12 ((𝐺𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺𝑢)) ⊆ (𝐹𝑡))
2514, 24syl5eqss 3633 . . . . . . . . . . 11 ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡))
2623, 25syl11 33 . . . . . . . . . 10 ((𝐹𝑡) ⊆ 𝑠 → ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2726reximdv 3015 . . . . . . . . 9 ((𝐹𝑡) ⊆ 𝑠 → (∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2827com12 32 . . . . . . . 8 (∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡 → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2928adantl 482 . . . . . . 7 ((𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
3022, 29syl6bi 243 . . . . . 6 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3130rexlimdv 3028 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
3220, 31impbid 202 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 ↔ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
3332anbi2d 739 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠) ↔ (𝑠𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)))
34 simpl1 1062 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑋𝑉)
35 fco 6017 . . . . 5 ((𝐹:𝑌𝑋𝐺:𝑍𝑌) → (𝐹𝐺):𝑍𝑋)
3635adantl 482 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝐹𝐺):𝑍𝑋)
37 elfm 21656 . . . 4 ((𝑋𝑉𝐵 ∈ (fBas‘𝑍) ∧ (𝐹𝐺):𝑍𝑋) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3834, 1, 36, 37syl3anc 1323 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
39 fmfil 21653 . . . . . 6 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
405, 1, 6, 39syl3anc 1323 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41 filfbas 21557 . . . . 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 793 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐹:𝑌𝑋)
44 elfm 21656 . . . 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 1323 . . 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 300 . 2 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ 𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵))))
4746eqrdv 2624 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wrex 2913  wss 3560  cima 5082  ccom 5083  wf 5846  cfv 5850  (class class class)co 6605  fBascfbas 19648  filGencfg 19649  Filcfil 21554   FilMap cfm 21642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-fbas 19657  df-fg 19658  df-fil 21555  df-fm 21647
This theorem is referenced by:  ufldom  21671  flfcnp  21713
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