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Theorem fmfg 22276
Description: The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
elfm2.l 𝐿 = (𝑌filGen𝐵)
Assertion
Ref Expression
fmfg ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))

Proof of Theorem fmfg
Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfm2.l . . . 4 𝐿 = (𝑌filGen𝐵)
21elfm2 22275 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3 fgcl 22205 . . . . . 6 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
41, 3syl5eqel 2872 . . . . 5 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
5 filfbas 22175 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
64, 5syl 17 . . . 4 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
7 elfm 22274 . . . 4 ((𝑋𝐶𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
86, 7syl3an2 1145 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
92, 8bitr4d 274 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))
109eqrdv 2778 1 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wrex 3091  wss 3831  cima 5414  wf 6189  cfv 6193  (class class class)co 6982  fBascfbas 20250  filGencfg 20251  Filcfil 22172   FilMap cfm 22260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-ov 6985  df-oprab 6986  df-mpo 6987  df-fbas 20259  df-fg 20260  df-fil 22173  df-fm 22265
This theorem is referenced by:  fmfnfm  22285  cmetcaulem  23609
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