Theorem fmss 23890

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 1193	. . . 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 1192	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝑋 ∈ 𝐴)
4		eqid 2736	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
5	4	fbasrn 23828	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
6	1, 2, 3, 5	syl3anc 1373	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
7		simpl3 1194	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → 𝐶 ∈ (fBas‘𝑌))
8		eqid 2736	. . . . 5 ⊢ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
9	8	fbasrn 23828	. . . 4 ⊢ ((𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
10	7, 2, 3, 9	syl3anc 1373	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
11		resmpt 5996	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 → ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
12	11	ad2antll 729	. . . . 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 5960	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
14	12, 13	eqsstrrdi 3979	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
15		rnss 5888	. . . 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 23817	. . 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 1373	. 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 23887	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
20	3, 1, 2, 19	syl3anc 1373	. 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 23887	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐶 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐶) = (𝑋filGenran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))))
22	3, 7, 2, 21	syl3anc 1373	. 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 3989	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 1086   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901   ↦ cmpt 5179  ran crn 5625   ↾ cres 5626   “ cima 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  fBascfbas 21297  filGencfg 21298   FilMap cfm 23877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fbas 21306  df-fg 21307  df-fm 23882

This theorem is referenced by:  ufldom  23906  cnpfcfi  23984