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Theorem elfm3 22166
Description: An alternate formulation of elementhood in a mapping filter that requires 𝐹 to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
elfm2.l 𝐿 = (𝑌filGen𝐵)
Assertion
Ref Expression
elfm3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝑋   𝑥,𝐴   𝑥,𝐿   𝑥,𝑌

Proof of Theorem elfm3
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foima 6373 . . . 4 (𝐹:𝑌onto𝑋 → (𝐹𝑌) = 𝑋)
21adantl 475 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) = 𝑋)
3 fofun 6369 . . . 4 (𝐹:𝑌onto𝑋 → Fun 𝐹)
4 elfvdm 6480 . . . 4 (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5 funimaexg 6222 . . . 4 ((Fun 𝐹𝑌 ∈ dom fBas) → (𝐹𝑌) ∈ V)
63, 4, 5syl2anr 590 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) ∈ V)
72, 6eqeltrrd 2860 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝑋 ∈ V)
8 fof 6368 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
9 elfm2.l . . . . . 6 𝐿 = (𝑌filGen𝐵)
109elfm2 22164 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
118, 10syl3an3 1166 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
12 fgcl 22094 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
139, 12syl5eqel 2863 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14133ad2ant2 1125 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝐿 ∈ (Fil‘𝑌))
1514ad2antrr 716 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌))
16 simprl 761 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦𝐿)
17 cnvimass 5741 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ dom 𝐹
18 fofn 6370 . . . . . . . . . . . . 13 (𝐹:𝑌onto𝑋𝐹 Fn 𝑌)
19 fndm 6237 . . . . . . . . . . . . 13 (𝐹 Fn 𝑌 → dom 𝐹 = 𝑌)
2018, 19syl 17 . . . . . . . . . . . 12 (𝐹:𝑌onto𝑋 → dom 𝐹 = 𝑌)
2117, 20syl5sseq 3872 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → (𝐹𝐴) ⊆ 𝑌)
22213ad2ant3 1126 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝐴) ⊆ 𝑌)
2322ad2antrr 716 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ⊆ 𝑌)
2433ad2ant3 1126 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → Fun 𝐹)
2524ad2antrr 716 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → Fun 𝐹)
269eleq2i 2851 . . . . . . . . . . . . . . 15 (𝑦𝐿𝑦 ∈ (𝑌filGen𝐵))
27 elfg 22087 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
28273ad2ant2 1125 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
2928adantr 474 . . . . . . . . . . . . . . 15 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3026, 29syl5bb 275 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦𝐿 ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3130simprbda 494 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦𝑌)
32 sseq2 3846 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹𝑦𝑌))
3332biimpar 471 . . . . . . . . . . . . . . . 16 ((dom 𝐹 = 𝑌𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3420, 33sylan 575 . . . . . . . . . . . . . . 15 ((𝐹:𝑌onto𝑋𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
35343ad2antl3 1195 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3635adantlr 705 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3731, 36syldan 585 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦 ⊆ dom 𝐹)
38 funimass3 6598 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ⊆ dom 𝐹) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
3925, 37, 38syl2anc 579 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4039biimpd 221 . . . . . . . . . 10 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4140impr 448 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (𝐹𝐴))
42 filss 22069 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦𝐿 ∧ (𝐹𝐴) ⊆ 𝑌𝑦 ⊆ (𝐹𝐴))) → (𝐹𝐴) ∈ 𝐿)
4315, 16, 23, 41, 42syl13anc 1440 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ∈ 𝐿)
44 foimacnv 6410 . . . . . . . . . . 11 ((𝐹:𝑌onto𝑋𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
4544eqcomd 2784 . . . . . . . . . 10 ((𝐹:𝑌onto𝑋𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
46453ad2antl3 1195 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
4746adantr 474 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (𝐹𝐴)))
48 imaeq2 5718 . . . . . . . . 9 (𝑥 = (𝐹𝐴) → (𝐹𝑥) = (𝐹 “ (𝐹𝐴)))
4948rspceeqv 3529 . . . . . . . 8 (((𝐹𝐴) ∈ 𝐿𝐴 = (𝐹 “ (𝐹𝐴))) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5043, 47, 49syl2anc 579 . . . . . . 7 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5150rexlimdvaa 3214 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴 → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
5251expimpd 447 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
53 simprr 763 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴 = (𝐹𝑥))
54 imassrn 5733 . . . . . . . . 9 (𝐹𝑥) ⊆ ran 𝐹
55 forn 6371 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → ran 𝐹 = 𝑋)
56553ad2ant3 1126 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ran 𝐹 = 𝑋)
5756adantr 474 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ran 𝐹 = 𝑋)
5854, 57syl5sseq 3872 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐹𝑥) ⊆ 𝑋)
5953, 58eqsstrd 3858 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴𝑋)
60 eqimss2 3877 . . . . . . . . 9 (𝐴 = (𝐹𝑥) → (𝐹𝑥) ⊆ 𝐴)
61 imaeq2 5718 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6261sseq1d 3851 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ⊆ 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
6362rspcev 3511 . . . . . . . . 9 ((𝑥𝐿 ∧ (𝐹𝑥) ⊆ 𝐴) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6460, 63sylan2 586 . . . . . . . 8 ((𝑥𝐿𝐴 = (𝐹𝑥)) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6564adantl 475 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6659, 65jca 507 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴))
6766rexlimdvaa 3214 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (∃𝑥𝐿 𝐴 = (𝐹𝑥) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
6852, 67impbid 204 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
6911, 68bitrd 271 . . 3 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
70693coml 1118 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
717, 70mpd3an3 1535 1 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wrex 3091  Vcvv 3398  wss 3792  ccnv 5356  dom cdm 5357  ran crn 5358  cima 5360  Fun wfun 6131   Fn wfn 6132  wf 6133  ontowfo 6135  cfv 6137  (class class class)co 6924  fBascfbas 20134  filGencfg 20135  Filcfil 22061   FilMap cfm 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-fbas 20143  df-fg 20144  df-fil 22062  df-fm 22154
This theorem is referenced by:  fmid  22176
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