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Theorem fmss 23954
Description: A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
fmss (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶))

Proof of Theorem fmss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl2 1193 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2 simprl 771 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐹:𝑌𝑋)
3 simpl1 1192 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝑋𝐴)
4 eqid 2737 . . . . 5 ran (𝑦𝐵 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦))
54fbasrn 23892 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
61, 2, 3, 5syl3anc 1373 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
7 simpl3 1194 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8 eqid 2737 . . . . 5 ran (𝑦𝐶 ↦ (𝐹𝑦)) = ran (𝑦𝐶 ↦ (𝐹𝑦))
98fbasrn 23892 . . . 4 ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
107, 2, 3, 9syl3anc 1373 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
11 resmpt 6055 . . . . . 6 (𝐵𝐶 → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
1211ad2antll 729 . . . . 5 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
13 resss 6019 . . . . 5 ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) ⊆ (𝑦𝐶 ↦ (𝐹𝑦))
1412, 13eqsstrrdi 4029 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ (𝑦𝐶 ↦ (𝐹𝑦)))
15 rnss 5950 . . . 4 ((𝑦𝐵 ↦ (𝐹𝑦)) ⊆ (𝑦𝐶 ↦ (𝐹𝑦)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ ran (𝑦𝐶 ↦ (𝐹𝑦)))
1614, 15syl 17 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ ran (𝑦𝐶 ↦ (𝐹𝑦)))
17 fgss 23881 . . 3 ((ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋) ∧ ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋) ∧ ran (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ ran (𝑦𝐶 ↦ (𝐹𝑦))) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ⊆ (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
186, 10, 16, 17syl3anc 1373 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ⊆ (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
19 fmval 23951 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
203, 1, 2, 19syl3anc 1373 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
21 fmval 23951 . . 3 ((𝑋𝐴𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
223, 7, 2, 21syl3anc 1373 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
2318, 20, 223sstr4d 4039 1 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  cmpt 5225  ran crn 5686  cres 5687  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353   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-fm 23946
This theorem is referenced by:  ufldom  23970  cnpfcfi  24048
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