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Theorem elfm3 22550
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 6588 . . . 4 (𝐹:𝑌onto𝑋 → (𝐹𝑌) = 𝑋)
21adantl 484 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) = 𝑋)
3 fofun 6584 . . . 4 (𝐹:𝑌onto𝑋 → Fun 𝐹)
4 elfvdm 6695 . . . 4 (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5 funimaexg 6433 . . . 4 ((Fun 𝐹𝑌 ∈ dom fBas) → (𝐹𝑌) ∈ V)
63, 4, 5syl2anr 598 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) ∈ V)
72, 6eqeltrrd 2912 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝑋 ∈ V)
8 fof 6583 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
9 elfm2.l . . . . . 6 𝐿 = (𝑌filGen𝐵)
109elfm2 22548 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
118, 10syl3an3 1160 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
12 fgcl 22478 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
139, 12eqeltrid 2915 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14133ad2ant2 1129 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝐿 ∈ (Fil‘𝑌))
1514ad2antrr 724 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌))
16 simprl 769 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦𝐿)
17 cnvimass 5942 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ dom 𝐹
18 fofn 6585 . . . . . . . . . . . . 13 (𝐹:𝑌onto𝑋𝐹 Fn 𝑌)
19 fndm 6448 . . . . . . . . . . . . 13 (𝐹 Fn 𝑌 → dom 𝐹 = 𝑌)
2018, 19syl 17 . . . . . . . . . . . 12 (𝐹:𝑌onto𝑋 → dom 𝐹 = 𝑌)
2117, 20sseqtrid 4017 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → (𝐹𝐴) ⊆ 𝑌)
22213ad2ant3 1130 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝐴) ⊆ 𝑌)
2322ad2antrr 724 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ⊆ 𝑌)
2433ad2ant3 1130 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → Fun 𝐹)
2524ad2antrr 724 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → Fun 𝐹)
269eleq2i 2902 . . . . . . . . . . . . . . 15 (𝑦𝐿𝑦 ∈ (𝑌filGen𝐵))
27 elfg 22471 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
28273ad2ant2 1129 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
2928adantr 483 . . . . . . . . . . . . . . 15 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3026, 29syl5bb 285 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦𝐿 ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3130simprbda 501 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦𝑌)
32 sseq2 3991 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹𝑦𝑌))
3332biimpar 480 . . . . . . . . . . . . . . . 16 ((dom 𝐹 = 𝑌𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3420, 33sylan 582 . . . . . . . . . . . . . . 15 ((𝐹:𝑌onto𝑋𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
35343ad2antl3 1182 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3635adantlr 713 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3731, 36syldan 593 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦 ⊆ dom 𝐹)
38 funimass3 6817 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ⊆ dom 𝐹) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
3925, 37, 38syl2anc 586 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4039biimpd 231 . . . . . . . . . 10 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4140impr 457 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (𝐹𝐴))
42 filss 22453 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦𝐿 ∧ (𝐹𝐴) ⊆ 𝑌𝑦 ⊆ (𝐹𝐴))) → (𝐹𝐴) ∈ 𝐿)
4315, 16, 23, 41, 42syl13anc 1367 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ∈ 𝐿)
44 foimacnv 6625 . . . . . . . . . . 11 ((𝐹:𝑌onto𝑋𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
4544eqcomd 2825 . . . . . . . . . 10 ((𝐹:𝑌onto𝑋𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
46453ad2antl3 1182 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
4746adantr 483 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (𝐹𝐴)))
48 imaeq2 5918 . . . . . . . . 9 (𝑥 = (𝐹𝐴) → (𝐹𝑥) = (𝐹 “ (𝐹𝐴)))
4948rspceeqv 3636 . . . . . . . 8 (((𝐹𝐴) ∈ 𝐿𝐴 = (𝐹 “ (𝐹𝐴))) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5043, 47, 49syl2anc 586 . . . . . . 7 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5150rexlimdvaa 3283 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴 → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
5251expimpd 456 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
53 simprr 771 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴 = (𝐹𝑥))
54 imassrn 5933 . . . . . . . . 9 (𝐹𝑥) ⊆ ran 𝐹
55 forn 6586 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → ran 𝐹 = 𝑋)
56553ad2ant3 1130 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ran 𝐹 = 𝑋)
5756adantr 483 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ran 𝐹 = 𝑋)
5854, 57sseqtrid 4017 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐹𝑥) ⊆ 𝑋)
5953, 58eqsstrd 4003 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴𝑋)
60 eqimss2 4022 . . . . . . . . 9 (𝐴 = (𝐹𝑥) → (𝐹𝑥) ⊆ 𝐴)
61 imaeq2 5918 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6261sseq1d 3996 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ⊆ 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
6362rspcev 3621 . . . . . . . . 9 ((𝑥𝐿 ∧ (𝐹𝑥) ⊆ 𝐴) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6460, 63sylan2 594 . . . . . . . 8 ((𝑥𝐿𝐴 = (𝐹𝑥)) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6564adantl 484 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6659, 65jca 514 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴))
6766rexlimdvaa 3283 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (∃𝑥𝐿 𝐴 = (𝐹𝑥) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
6852, 67impbid 214 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
6911, 68bitrd 281 . . 3 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
70693coml 1122 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
717, 70mpd3an3 1456 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 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wrex 3137  Vcvv 3493  wss 3934  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  Fun wfun 6342   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  (class class class)co 7148  fBascfbas 20525  filGencfg 20526  Filcfil 22445   FilMap cfm 22533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-fbas 20534  df-fg 20535  df-fil 22446  df-fm 22538
This theorem is referenced by:  fmid  22560
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