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Theorem elfm3 21564
 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 6033 . . . 4 (𝐹:𝑌onto𝑋 → (𝐹𝑌) = 𝑋)
21adantl 481 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) = 𝑋)
3 fofun 6029 . . . 4 (𝐹:𝑌onto𝑋 → Fun 𝐹)
4 elfvdm 6130 . . . 4 (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5 funimaexg 5889 . . . 4 ((Fun 𝐹𝑌 ∈ dom fBas) → (𝐹𝑌) ∈ V)
63, 4, 5syl2anr 494 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) ∈ V)
72, 6eqeltrrd 2689 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝑋 ∈ V)
8 fof 6028 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
9 elfm2.l . . . . . 6 𝐿 = (𝑌filGen𝐵)
109elfm2 21562 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
118, 10syl3an3 1353 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
12 fgcl 21492 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
139, 12syl5eqel 2692 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14133ad2ant2 1076 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝐿 ∈ (Fil‘𝑌))
1514ad2antrr 758 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐿 ∈ (Fil‘𝑌))
16 simprl 790 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦𝐿)
17 cnvimass 5404 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ dom 𝐹
18 fofn 6030 . . . . . . . . . . . . 13 (𝐹:𝑌onto𝑋𝐹 Fn 𝑌)
19 fndm 5904 . . . . . . . . . . . . 13 (𝐹 Fn 𝑌 → dom 𝐹 = 𝑌)
2018, 19syl 17 . . . . . . . . . . . 12 (𝐹:𝑌onto𝑋 → dom 𝐹 = 𝑌)
2117, 20syl5sseq 3616 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → (𝐹𝐴) ⊆ 𝑌)
22213ad2ant3 1077 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝐴) ⊆ 𝑌)
2322ad2antrr 758 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ⊆ 𝑌)
2433ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → Fun 𝐹)
2524ad2antrr 758 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → Fun 𝐹)
269eleq2i 2680 . . . . . . . . . . . . . . 15 (𝑦𝐿𝑦 ∈ (𝑌filGen𝐵))
27 elfg 21485 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
28273ad2ant2 1076 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
2928adantr 480 . . . . . . . . . . . . . . 15 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3026, 29syl5bb 271 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦𝐿 ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3130simprbda 651 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦𝑌)
32 sseq2 3590 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹𝑦𝑌))
3332biimpar 501 . . . . . . . . . . . . . . . 16 ((dom 𝐹 = 𝑌𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3420, 33sylan 487 . . . . . . . . . . . . . . 15 ((𝐹:𝑌onto𝑋𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
35343ad2antl3 1218 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3635adantlr 747 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3731, 36syldan 486 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦 ⊆ dom 𝐹)
38 funimass3 6241 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ⊆ dom 𝐹) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
3925, 37, 38syl2anc 691 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4039biimpd 218 . . . . . . . . . 10 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4140impr 647 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (𝐹𝐴))
42 filss 21467 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦𝐿 ∧ (𝐹𝐴) ⊆ 𝑌𝑦 ⊆ (𝐹𝐴))) → (𝐹𝐴) ∈ 𝐿)
4315, 16, 23, 41, 42syl13anc 1320 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ∈ 𝐿)
44 foimacnv 6067 . . . . . . . . . . 11 ((𝐹:𝑌onto𝑋𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
4544eqcomd 2616 . . . . . . . . . 10 ((𝐹:𝑌onto𝑋𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
46453ad2antl3 1218 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
4746adantr 480 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (𝐹𝐴)))
48 imaeq2 5381 . . . . . . . . . 10 (𝑥 = (𝐹𝐴) → (𝐹𝑥) = (𝐹 “ (𝐹𝐴)))
4948eqeq2d 2620 . . . . . . . . 9 (𝑥 = (𝐹𝐴) → (𝐴 = (𝐹𝑥) ↔ 𝐴 = (𝐹 “ (𝐹𝐴))))
5049rspcev 3282 . . . . . . . 8 (((𝐹𝐴) ∈ 𝐿𝐴 = (𝐹 “ (𝐹𝐴))) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5143, 47, 50syl2anc 691 . . . . . . 7 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5251rexlimdvaa 3014 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴 → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
5352expimpd 627 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
54 simprr 792 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴 = (𝐹𝑥))
55 imassrn 5396 . . . . . . . . 9 (𝐹𝑥) ⊆ ran 𝐹
56 forn 6031 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → ran 𝐹 = 𝑋)
57563ad2ant3 1077 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ran 𝐹 = 𝑋)
5857adantr 480 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ran 𝐹 = 𝑋)
5955, 58syl5sseq 3616 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐹𝑥) ⊆ 𝑋)
6054, 59eqsstrd 3602 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴𝑋)
61 eqimss2 3621 . . . . . . . . 9 (𝐴 = (𝐹𝑥) → (𝐹𝑥) ⊆ 𝐴)
62 imaeq2 5381 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6362sseq1d 3595 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ⊆ 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
6463rspcev 3282 . . . . . . . . 9 ((𝑥𝐿 ∧ (𝐹𝑥) ⊆ 𝐴) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6561, 64sylan2 490 . . . . . . . 8 ((𝑥𝐿𝐴 = (𝐹𝑥)) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6665adantl 481 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6760, 66jca 553 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴))
6867rexlimdvaa 3014 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (∃𝑥𝐿 𝐴 = (𝐹𝑥) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
6953, 68impbid 201 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
7011, 69bitrd 267 . . 3 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
71703coml 1264 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
727, 71mpd3an3 1417 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  fBascfbas 19555  filGencfg 19556  Filcfil 21459   FilMap cfm 21547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fg 19565  df-fil 21460  df-fm 21552 This theorem is referenced by:  fmid  21574
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