Theorem fmss 23975

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 1192	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐵 ∈ (fBas‘𝑌))
2		simprl 770	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐹:𝑌⟶𝑋)
3		simpl1 1191	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝑋 ∈ 𝐴)
4		eqid 2740	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
5	4	fbasrn 23913	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
6	1, 2, 3, 5	syl3anc 1371	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
7		simpl3 1193	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8		eqid 2740	. . . . 5 ⊢ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
9	8	fbasrn 23913	. . . 4 ⊢ ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
10	7, 2, 3, 9	syl3anc 1371	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
11		resmpt 6066	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 → ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
12	11	ad2antll 728	. . . . 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 6031	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
14	12, 13	eqsstrrdi 4064	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
15		rnss 5964	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
16	14, 15	syl 17	. . 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 23902	. . 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 1371	. 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 23972	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
20	3, 1, 2, 19	syl3anc 1371	. 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 23972	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))))
22	3, 7, 2, 21	syl3anc 1371	. 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 4056	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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976   ↦ cmpt 5249  ran crn 5701   ↾ cres 5702   “ cima 5703  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  fBascfbas 21375  filGencfg 21376   FilMap cfm 23962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-fbas 21384  df-fg 21385  df-fm 23967

This theorem is referenced by:  ufldom  23991  cnpfcfi  24069