Theorem fmss 24280

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 2763	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
5	4	fbasrn 24218	. . . 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 24218	. . . 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 6035	. . . . . 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 5997	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)) ↾ 𝐵) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦))
14	12, 13	eqsstrrdi 3958	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ⊆ (𝑦 ∈ 𝐶 ↦ (𝐹 “ 𝑦)))
15		rnss 5926	. . . 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 24207	. . 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 24277	. . 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 24277	. . 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 3968	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 1569   ∈ wcel 2146   ⊆ wss 3900   ↦ cmpt 5189  ran crn 5656   ↾ cres 5657   “ cima 5658  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430  fBascfbas 21681  filGencfg 21682   FilMap cfm 24267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  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 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-fbas 21690  df-fg 21691  df-fm 24272

This theorem is used by:  ufldom  24296  cnpfcfi  24374