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Theorem fmco 22173
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 1203 . . . . . . . . . . 11 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2 ssfg 22084 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑍) → 𝐵 ⊆ (𝑍filGen𝐵))
31, 2syl 17 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐵 ⊆ (𝑍filGen𝐵))
43sseld 3820 . . . . . . . . 9 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑢𝐵𝑢 ∈ (𝑍filGen𝐵)))
5 simpl2 1201 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑌𝑊)
6 simprr 763 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐺:𝑍𝑌)
7 eqid 2778 . . . . . . . . . . . 12 (𝑍filGen𝐵) = (𝑍filGen𝐵)
87imaelfm 22163 . . . . . . . . . . 11 (((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
98ex 403 . . . . . . . . . 10 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
105, 1, 6, 9syl3anc 1439 . . . . . . . . 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 397 . . . . . . 7 ((((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) ∧ 𝑢𝐵) → (𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13 imaeq2 5716 . . . . . . . . . . 11 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = (𝐹 “ (𝐺𝑢)))
14 imaco 5894 . . . . . . . . . . 11 ((𝐹𝐺) “ 𝑢) = (𝐹 “ (𝐺𝑢))
1513, 14syl6eqr 2832 . . . . . . . . . 10 (𝑡 = (𝐺𝑢) → (𝐹𝑡) = ((𝐹𝐺) “ 𝑢))
1615sseq1d 3851 . . . . . . . . 9 (𝑡 = (𝐺𝑢) → ((𝐹𝑡) ⊆ 𝑠 ↔ ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
1716rspcev 3511 . . . . . . . 8 (((𝐺𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)
1817ex 403 . . . . . . 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 3213 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
21 elfm 22159 . . . . . . . 8 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
225, 1, 6, 21syl3anc 1439 . . . . . . 7 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡)))
23 sstr2 3828 . . . . . . . . . . 11 (((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
24 imass2 5755 . . . . . . . . . . . 12 ((𝐺𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺𝑢)) ⊆ (𝐹𝑡))
2514, 24syl5eqss 3868 . . . . . . . . . . 11 ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ (𝐹𝑡))
2623, 25syl11 33 . . . . . . . . . 10 ((𝐹𝑡) ⊆ 𝑠 → ((𝐺𝑢) ⊆ 𝑡 → ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2726reximdv 3197 . . . . . . . . 9 ((𝐹𝑡) ⊆ 𝑠 → (∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2827com12 32 . . . . . . . 8 (∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡 → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
2928adantl 475 . . . . . . 7 ((𝑡𝑌 ∧ ∃𝑢𝐵 (𝐺𝑢) ⊆ 𝑡) → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
3022, 29syl6bi 245 . . . . . 6 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) → ((𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3130rexlimdv 3212 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠 → ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠))
3220, 31impbid 204 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠 ↔ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠))
3332anbi2d 622 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠) ↔ (𝑠𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹𝑡) ⊆ 𝑠)))
34 simpl1 1199 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝑋𝑉)
35 fco 6308 . . . . 5 ((𝐹:𝑌𝑋𝐺:𝑍𝑌) → (𝐹𝐺):𝑍𝑋)
3635adantl 475 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝐹𝐺):𝑍𝑋)
37 elfm 22159 . . . 4 ((𝑋𝑉𝐵 ∈ (fBas‘𝑍) ∧ (𝐹𝐺):𝑍𝑋) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
3834, 1, 36, 37syl3anc 1439 . . 3 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ (𝑠𝑋 ∧ ∃𝑢𝐵 ((𝐹𝐺) “ 𝑢) ⊆ 𝑠)))
39 fmfil 22156 . . . . . 6 ((𝑌𝑊𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
405, 1, 6, 39syl3anc 1439 . . . . 5 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41 filfbas 22060 . . . . 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 761 . . . 4 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → 𝐹:𝑌𝑋)
44 elfm 22159 . . . 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 1439 . . 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 303 . 2 (((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹𝐺))‘𝐵) ↔ 𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵))))
4746eqrdv 2776 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wrex 3091  wss 3792  cima 5358  ccom 5359  wf 6131  cfv 6135  (class class class)co 6922  fBascfbas 20130  filGencfg 20131  Filcfil 22057   FilMap cfm 22145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-fbas 20139  df-fg 20140  df-fil 22058  df-fm 22150
This theorem is referenced by:  ufldom  22174  flfcnp  22216
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