Theorem fmfil 22554

Description: A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
fmfil	⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋))

Proof of Theorem fmfil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmval 22553	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
2		eqid 2823	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
3	2	fbasrn 22494	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ 𝐴) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
4	3	3comr 1121	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
5		fgcl 22488	. . 3 ⊢ (ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋) → (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (Fil‘𝑋))
6	4, 5	syl 17	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (Fil‘𝑋))
7	1, 6	eqeltrd 2915	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ∧ w3a 1083   ∈ wcel 2114   ↦ cmpt 5148  ran crn 5558   “ cima 5560  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  fBascfbas 20535  filGencfg 20536  Filcfil 22455   FilMap cfm 22543

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-fbas 20544  df-fg 20545  df-fil 22456  df-fm 22548

This theorem is referenced by:  fmf  22555  fmufil  22569  fmco  22571  ufldom  22572  flfnei  22601  isflf  22603  flfcnp  22614  isfcf  22644  cnpfcfi  22650  cnpfcf  22651  cnextucn  22914