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Theorem fmss 22120
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 1248 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2 simprl 787 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐹:𝑌𝑋)
3 simpl1 1246 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝑋𝐴)
4 eqid 2825 . . . . 5 ran (𝑦𝐵 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦))
54fbasrn 22058 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
61, 2, 3, 5syl3anc 1494 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
7 simpl3 1250 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8 eqid 2825 . . . . 5 ran (𝑦𝐶 ↦ (𝐹𝑦)) = ran (𝑦𝐶 ↦ (𝐹𝑦))
98fbasrn 22058 . . . 4 ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
107, 2, 3, 9syl3anc 1494 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
11 resmpt 5686 . . . . . 6 (𝐵𝐶 → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
1211ad2antll 720 . . . . 5 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
13 resss 5658 . . . . 5 ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) ⊆ (𝑦𝐶 ↦ (𝐹𝑦))
1412, 13syl6eqssr 3881 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ (𝑦𝐶 ↦ (𝐹𝑦)))
15 rnss 5586 . . . 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 22047 . . 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 1494 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ⊆ (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
19 fmval 22117 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
203, 1, 2, 19syl3anc 1494 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
21 fmval 22117 . . 3 ((𝑋𝐴𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
223, 7, 2, 21syl3anc 1494 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
2318, 20, 223sstr4d 3873 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 386  w3a 1111   = wceq 1656  wcel 2164  wss 3798  cmpt 4952  ran crn 5343  cres 5344  cima 5345  wf 6119  cfv 6123  (class class class)co 6905  fBascfbas 20094  filGencfg 20095   FilMap cfm 22107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-fbas 20103  df-fg 20104  df-fm 22112
This theorem is referenced by:  ufldom  22136  cnpfcfi  22214
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