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Theorem fmss 22558
 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 1189 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2 simprl 770 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐹:𝑌𝑋)
3 simpl1 1188 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝑋𝐴)
4 eqid 2798 . . . . 5 ran (𝑦𝐵 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦))
54fbasrn 22496 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
61, 2, 3, 5syl3anc 1368 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
7 simpl3 1190 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8 eqid 2798 . . . . 5 ran (𝑦𝐶 ↦ (𝐹𝑦)) = ran (𝑦𝐶 ↦ (𝐹𝑦))
98fbasrn 22496 . . . 4 ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
107, 2, 3, 9syl3anc 1368 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
11 resmpt 5872 . . . . . 6 (𝐵𝐶 → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
1211ad2antll 728 . . . . 5 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
13 resss 5843 . . . . 5 ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) ⊆ (𝑦𝐶 ↦ (𝐹𝑦))
1412, 13eqsstrrdi 3970 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ (𝑦𝐶 ↦ (𝐹𝑦)))
15 rnss 5773 . . . 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 22485 . . 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 1368 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ⊆ (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
19 fmval 22555 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
203, 1, 2, 19syl3anc 1368 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
21 fmval 22555 . . 3 ((𝑋𝐴𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
223, 7, 2, 21syl3anc 1368 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
2318, 20, 223sstr4d 3962 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  fBascfbas 20082  filGencfg 20083   FilMap cfm 22545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-fbas 20091  df-fg 20092  df-fm 22550 This theorem is referenced by:  ufldom  22574  cnpfcfi  22652
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