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Theorem elfm 23098
Description: An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
elfm ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem elfm
Dummy variables 𝑡 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmval 23094 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))))
21eleq2d 2824 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡)))))
3 eqid 2738 . . . . 5 ran (𝑡𝐵 ↦ (𝐹𝑡)) = ran (𝑡𝐵 ↦ (𝐹𝑡))
43fbasrn 23035 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐶) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
543comr 1124 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
6 elfg 23022 . . 3 (ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))) ↔ (𝐴𝑋 ∧ ∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴)))
75, 6syl 17 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))) ↔ (𝐴𝑋 ∧ ∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴)))
8 simpr 485 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑥𝐵)
9 eqid 2738 . . . . . 6 (𝐹𝑥) = (𝐹𝑥)
10 imaeq2 5965 . . . . . . 7 (𝑡 = 𝑥 → (𝐹𝑡) = (𝐹𝑥))
1110rspceeqv 3575 . . . . . 6 ((𝑥𝐵 ∧ (𝐹𝑥) = (𝐹𝑥)) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
128, 9, 11sylancl 586 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
13 simpl1 1190 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑋𝐶)
14 imassrn 5980 . . . . . . . 8 (𝐹𝑥) ⊆ ran 𝐹
15 frn 6607 . . . . . . . . . 10 (𝐹:𝑌𝑋 → ran 𝐹𝑋)
16153ad2ant3 1134 . . . . . . . . 9 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran 𝐹𝑋)
1716adantr 481 . . . . . . . 8 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ran 𝐹𝑋)
1814, 17sstrid 3932 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑋)
1913, 18ssexd 5248 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ V)
20 eqid 2738 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑡𝐵 ↦ (𝐹𝑡))
2120elrnmpt 5865 . . . . . 6 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡)))
2219, 21syl 17 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ((𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡)))
2312, 22mpbird 256 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)))
2410cbvmptv 5187 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑥𝐵 ↦ (𝐹𝑥))
2524elrnmpt 5865 . . . . . 6 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
2625ibi 266 . . . . 5 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
2726adantl 482 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
28 simpr 485 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
2928sseq1d 3952 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → (𝑦𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
3023, 27, 29rexxfrd 5332 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴 ↔ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴))
3130anbi2d 629 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴) ↔ (𝐴𝑋 ∧ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴)))
322, 7, 313bitrd 305 1 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  wss 3887  cmpt 5157  ran crn 5590  cima 5592  wf 6429  cfv 6433  (class class class)co 7275  fBascfbas 20585  filGencfg 20586   FilMap cfm 23084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-fbas 20594  df-fg 20595  df-fm 23089
This theorem is referenced by:  elfm2  23099  fmfg  23100  rnelfm  23104  fmfnfmlem1  23105  fmfnfm  23109  fmco  23112  flfnei  23142  isflf  23144  isfcf  23185  filnetlem4  34570
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