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Theorem elfm 23971
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 23967 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))))
21eleq2d 2825 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡)))))
3 eqid 2735 . . . . 5 ran (𝑡𝐵 ↦ (𝐹𝑡)) = ran (𝑡𝐵 ↦ (𝐹𝑡))
43fbasrn 23908 . . . 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 23895 . . 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 484 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑥𝐵)
9 eqid 2735 . . . . . 6 (𝐹𝑥) = (𝐹𝑥)
10 imaeq2 6076 . . . . . . 7 (𝑡 = 𝑥 → (𝐹𝑡) = (𝐹𝑥))
1110rspceeqv 3645 . . . . . 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 6091 . . . . . . . 8 (𝐹𝑥) ⊆ ran 𝐹
15 frn 6744 . . . . . . . . . 10 (𝐹:𝑌𝑋 → ran 𝐹𝑋)
16153ad2ant3 1134 . . . . . . . . 9 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran 𝐹𝑋)
1716adantr 480 . . . . . . . 8 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ran 𝐹𝑋)
1814, 17sstrid 4007 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑋)
1913, 18ssexd 5330 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ V)
20 eqid 2735 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑡𝐵 ↦ (𝐹𝑡))
2120elrnmpt 5972 . . . . . 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 257 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)))
2410cbvmptv 5261 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑥𝐵 ↦ (𝐹𝑥))
2524elrnmpt 5972 . . . . . 6 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) ↔ ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
2625ibi 267 . . . . 5 (𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡)) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
2726adantl 481 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
28 simpr 484 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
2928sseq1d 4027 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → (𝑦𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
3023, 27, 29rexxfrd 5415 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (∃𝑦 ∈ ran (𝑡𝐵 ↦ (𝐹𝑡))𝑦𝐴 ↔ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴))
3130anbi2d 630 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963  cmpt 5231  ran crn 5690  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371   FilMap cfm 23957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fm 23962
This theorem is referenced by:  elfm2  23972  fmfg  23973  rnelfm  23977  fmfnfmlem1  23978  fmfnfm  23982  fmco  23985  flfnei  24015  isflf  24017  isfcf  24058  filnetlem4  36364
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