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Theorem fmss 22557
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 1188 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2 simprl 769 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐹:𝑌𝑋)
3 simpl1 1187 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝑋𝐴)
4 eqid 2824 . . . . 5 ran (𝑦𝐵 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦))
54fbasrn 22495 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
61, 2, 3, 5syl3anc 1367 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
7 simpl3 1189 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8 eqid 2824 . . . . 5 ran (𝑦𝐶 ↦ (𝐹𝑦)) = ran (𝑦𝐶 ↦ (𝐹𝑦))
98fbasrn 22495 . . . 4 ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
107, 2, 3, 9syl3anc 1367 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
11 resmpt 5908 . . . . . 6 (𝐵𝐶 → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
1211ad2antll 727 . . . . 5 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
13 resss 5881 . . . . 5 ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) ⊆ (𝑦𝐶 ↦ (𝐹𝑦))
1412, 13eqsstrrdi 4025 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ (𝑦𝐶 ↦ (𝐹𝑦)))
15 rnss 5812 . . . 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 22484 . . 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 1367 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ⊆ (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
19 fmval 22554 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
203, 1, 2, 19syl3anc 1367 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
21 fmval 22554 . . 3 ((𝑋𝐴𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
223, 7, 2, 21syl3anc 1367 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
2318, 20, 223sstr4d 4017 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 398  w3a 1083   = wceq 1536  wcel 2113  wss 3939  cmpt 5149  ran crn 5559  cres 5560  cima 5561  wf 6354  cfv 6358  (class class class)co 7159  fBascfbas 20536  filGencfg 20537   FilMap cfm 22544
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-fbas 20545  df-fg 20546  df-fm 22549
This theorem is referenced by:  ufldom  22573  cnpfcfi  22651
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