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Theorem elfm 22471
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 22467 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))))
21eleq2d 2903 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡)))))
3 eqid 2826 . . . . 5 ran (𝑡𝐵 ↦ (𝐹𝑡)) = ran (𝑡𝐵 ↦ (𝐹𝑡))
43fbasrn 22408 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐶) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
543comr 1119 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
6 elfg 22395 . . 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 2826 . . . . . 6 (𝐹𝑥) = (𝐹𝑥)
10 imaeq2 5923 . . . . . . 7 (𝑡 = 𝑥 → (𝐹𝑡) = (𝐹𝑥))
1110rspceeqv 3642 . . . . . 6 ((𝑥𝐵 ∧ (𝐹𝑥) = (𝐹𝑥)) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
128, 9, 11sylancl 586 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
13 simpl1 1185 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑋𝐶)
14 imassrn 5938 . . . . . . . 8 (𝐹𝑥) ⊆ ran 𝐹
15 frn 6517 . . . . . . . . . 10 (𝐹:𝑌𝑋 → ran 𝐹𝑋)
16153ad2ant3 1129 . . . . . . . . 9 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran 𝐹𝑋)
1716adantr 481 . . . . . . . 8 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ran 𝐹𝑋)
1814, 17sstrid 3982 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑋)
1913, 18ssexd 5225 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ V)
20 eqid 2826 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑡𝐵 ↦ (𝐹𝑡))
2120elrnmpt 5827 . . . . . 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 258 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)))
2410cbvmptv 5166 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑥𝐵 ↦ (𝐹𝑥))
2524elrnmpt 5827 . . . . . 6 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
2625ibi 268 . . . . 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 4002 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → (𝑦𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
3023, 27, 29rexxfrd 5306 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴 ↔ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴))
3130anbi2d 628 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴) ↔ (𝐴𝑋 ∧ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴)))
322, 7, 313bitrd 306 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wrex 3144  Vcvv 3500  wss 3940  cmpt 5143  ran crn 5555  cima 5557  wf 6348  cfv 6352  (class class class)co 7148  fBascfbas 20449  filGencfg 20450   FilMap cfm 22457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-fbas 20458  df-fg 20459  df-fm 22462
This theorem is referenced by:  elfm2  22472  fmfg  22473  rnelfm  22477  fmfnfmlem1  22478  fmfnfm  22482  fmco  22485  flfnei  22515  isflf  22517  isfcf  22558  filnetlem4  33613
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