Theorem fmss 24103

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	⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶))

Proof of Theorem fmss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1211	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2		simprl 782	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐹:𝑌⟶𝑋)
3		simpl1 1210	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝑋 ∈ 𝐴)
4		eqid 2763	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
5	4	fbasrn 24041	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
6	1, 2, 3, 5	syl3anc 1398	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
7		simpl3 1212	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8		eqid 2763	. . . . 5 ⊢ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
9	8	fbasrn 24041	. . . 4 ⊢ ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
10	7, 2, 3, 9	syl3anc 1398	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
11		resmpt 6039	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 → ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
12	11	ad2antll 741	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
13		resss 6000	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
14	12, 13	eqsstrrdi 3982	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
15		rnss 5929	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
16	14, 15	syl 18	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
17		fgss 24030	. . 3 ⊢ ((ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋) ∧ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋) ∧ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))) → (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ⊆ (𝑋filGenran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))))
18	6, 10, 16, 17	syl3anc 1398	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ⊆ (𝑋filGenran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))))
19		fmval 24100	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
20	3, 1, 2, 19	syl3anc 1398	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
21		fmval 24100	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))))
22	3, 7, 2, 21	syl3anc 1398	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))))
23	18, 20, 22	3sstr4d 3992	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905   ↦ cmpt 5192  ran crn 5662   ↾ cres 5663   “ cima 5664  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  fBascfbas 21510  filGencfg 21511   FilMap cfm 24090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-fbas 21519  df-fg 21520  df-fm 24095

This theorem is referenced by:  ufldom  24119  cnpfcfi  24197