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Theorem elfm 21691
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 21687 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))))
21eleq2d 2684 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡)))))
3 eqid 2621 . . . . 5 ran (𝑡𝐵 ↦ (𝐹𝑡)) = ran (𝑡𝐵 ↦ (𝐹𝑡))
43fbasrn 21628 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐶) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
543comr 1270 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
6 elfg 21615 . . 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 477 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑥𝐵)
9 eqid 2621 . . . . . 6 (𝐹𝑥) = (𝐹𝑥)
10 imaeq2 5431 . . . . . . . 8 (𝑡 = 𝑥 → (𝐹𝑡) = (𝐹𝑥))
1110eqeq2d 2631 . . . . . . 7 (𝑡 = 𝑥 → ((𝐹𝑥) = (𝐹𝑡) ↔ (𝐹𝑥) = (𝐹𝑥)))
1211rspcev 3299 . . . . . 6 ((𝑥𝐵 ∧ (𝐹𝑥) = (𝐹𝑥)) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
138, 9, 12sylancl 693 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
14 simpl1 1062 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑋𝐶)
15 imassrn 5446 . . . . . . . 8 (𝐹𝑥) ⊆ ran 𝐹
16 frn 6020 . . . . . . . . . 10 (𝐹:𝑌𝑋 → ran 𝐹𝑋)
17163ad2ant3 1082 . . . . . . . . 9 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran 𝐹𝑋)
1817adantr 481 . . . . . . . 8 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ran 𝐹𝑋)
1915, 18syl5ss 3599 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑋)
2014, 19ssexd 4775 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ V)
21 eqid 2621 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑡𝐵 ↦ (𝐹𝑡))
2221elrnmpt 5342 . . . . . 6 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡)))
2320, 22syl 17 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ((𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡)))
2413, 23mpbird 247 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)))
2510cbvmptv 4720 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑥𝐵 ↦ (𝐹𝑥))
2625elrnmpt 5342 . . . . . 6 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
2726ibi 256 . . . . 5 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
2827adantl 482 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
29 simpr 477 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
3029sseq1d 3617 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → (𝑦𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
3124, 28, 30rexxfrd 4851 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴 ↔ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴))
3231anbi2d 739 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴) ↔ (𝐴𝑋 ∧ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴)))
332, 7, 323bitrd 294 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2909  Vcvv 3190  wss 3560  cmpt 4683  ran crn 5085  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  fBascfbas 19674  filGencfg 19675   FilMap cfm 21677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-fbas 19683  df-fg 19684  df-fm 21682
This theorem is referenced by:  elfm2  21692  fmfg  21693  rnelfm  21697  fmfnfmlem1  21698  fmfnfm  21702  fmco  21705  flfnei  21735  isflf  21737  isfcf  21778  filnetlem4  32071
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