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Theorem elfm 23298
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 23294 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡))))
21eleq2d 2823 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝐴 ∈ (𝑋filGenran (𝑡𝐵 ↦ (𝐹𝑡)))))
3 eqid 2736 . . . . 5 ran (𝑡𝐵 ↦ (𝐹𝑡)) = ran (𝑡𝐵 ↦ (𝐹𝑡))
43fbasrn 23235 . . . 4 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋𝑋𝐶) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
543comr 1125 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran (𝑡𝐵 ↦ (𝐹𝑡)) ∈ (fBas‘𝑋))
6 elfg 23222 . . 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 2736 . . . . . 6 (𝐹𝑥) = (𝐹𝑥)
10 imaeq2 6009 . . . . . . 7 (𝑡 = 𝑥 → (𝐹𝑡) = (𝐹𝑥))
1110rspceeqv 3595 . . . . . 6 ((𝑥𝐵 ∧ (𝐹𝑥) = (𝐹𝑥)) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
128, 9, 11sylancl 586 . . . . 5 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ∃𝑡𝐵 (𝐹𝑥) = (𝐹𝑡))
13 simpl1 1191 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → 𝑋𝐶)
14 imassrn 6024 . . . . . . . 8 (𝐹𝑥) ⊆ ran 𝐹
15 frn 6675 . . . . . . . . . 10 (𝐹:𝑌𝑋 → ran 𝐹𝑋)
16153ad2ant3 1135 . . . . . . . . 9 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ran 𝐹𝑋)
1716adantr 481 . . . . . . . 8 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → ran 𝐹𝑋)
1814, 17sstrid 3955 . . . . . . 7 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ⊆ 𝑋)
1913, 18ssexd 5281 . . . . . 6 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ V)
20 eqid 2736 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑡𝐵 ↦ (𝐹𝑡))
2120elrnmpt 5911 . . . . . 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 5218 . . . . . . 7 (𝑡𝐵 ↦ (𝐹𝑡)) = (𝑥𝐵 ↦ (𝐹𝑥))
2524elrnmpt 5911 . . . . . 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 3975 . . . 4 (((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑦 = (𝐹𝑥)) → (𝑦𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
3023, 27, 29rexxfrd 5364 . . 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 304 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 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  wss 3910  cmpt 5188  ran crn 5634  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  fBascfbas 20784  filGencfg 20785   FilMap cfm 23284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-fbas 20793  df-fg 20794  df-fm 23289
This theorem is referenced by:  elfm2  23299  fmfg  23300  rnelfm  23304  fmfnfmlem1  23305  fmfnfm  23309  fmco  23312  flfnei  23342  isflf  23344  isfcf  23385  filnetlem4  34853
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