Theorem fmss 24175

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 783	. . . 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 2760	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
5	4	fbasrn 24113	. . . 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 2760	. . . . 5 ⊢ ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
9	8	fbasrn 24113	. . . 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 6033	. . . . . 6 ⊢ (𝐵 ⊆ 𝐶 → ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
12	11	ad2antll 742	. . . . 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 5994	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
14	12, 13	eqsstrrdi 3976	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
15		rnss 5923	. . . 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 24102	. . 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 24172	. . 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 24172	. . 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 3986	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657   “ cima 5658  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  fBascfbas 21576  filGencfg 21577   FilMap cfm 24162

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-fbas 21585  df-fg 21586  df-fm 24167

This theorem is used by:  ufldom  24191  cnpfcfi  24269