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Theorem fmss 21660
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 1063 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2 simprl 793 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐹:𝑌𝑋)
3 simpl1 1062 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝑋𝐴)
4 eqid 2621 . . . . 5 ran (𝑦𝐵 ↦ (𝐹𝑦)) = ran (𝑦𝐵 ↦ (𝐹𝑦))
54fbasrn 21598 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
61, 2, 3, 5syl3anc 1323 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐵 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
7 simpl3 1064 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8 eqid 2621 . . . . 5 ran (𝑦𝐶 ↦ (𝐹𝑦)) = ran (𝑦𝐶 ↦ (𝐹𝑦))
98fbasrn 21598 . . . 4 ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐴) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
107, 2, 3, 9syl3anc 1323 . . 3 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ran (𝑦𝐶 ↦ (𝐹𝑦)) ∈ (fBas‘𝑋))
11 resmpt 5408 . . . . . 6 (𝐵𝐶 → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
1211ad2antll 764 . . . . 5 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) = (𝑦𝐵 ↦ (𝐹𝑦)))
13 resss 5381 . . . . 5 ((𝑦𝐶 ↦ (𝐹𝑦)) ↾ 𝐵) ⊆ (𝑦𝐶 ↦ (𝐹𝑦))
1412, 13syl6eqssr 3635 . . . 4 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑦𝐵 ↦ (𝐹𝑦)) ⊆ (𝑦𝐶 ↦ (𝐹𝑦)))
15 rnss 5314 . . . 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 21587 . . 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 1323 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))) ⊆ (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
19 fmval 21657 . . 3 ((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
203, 1, 2, 19syl3anc 1323 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))
21 fmval 21657 . . 3 ((𝑋𝐴𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
223, 7, 2, 21syl3anc 1323 . 2 (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦𝐶 ↦ (𝐹𝑦))))
2318, 20, 223sstr4d 3627 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 384  w3a 1036   = wceq 1480  wcel 1987  wss 3555  cmpt 4673  ran crn 5075  cres 5076  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  fBascfbas 19653  filGencfg 19654   FilMap cfm 21647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-fbas 19662  df-fg 19663  df-fm 21652
This theorem is referenced by:  ufldom  21676  cnpfcfi  21754
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