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Theorem elfm3 23445
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 6807 . . . 4 (𝐹:𝑌onto𝑋 → (𝐹𝑌) = 𝑋)
21adantl 482 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) = 𝑋)
3 fofun 6803 . . . 4 (𝐹:𝑌onto𝑋 → Fun 𝐹)
4 elfvdm 6925 . . . 4 (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5 funimaexg 6631 . . . 4 ((Fun 𝐹𝑌 ∈ dom fBas) → (𝐹𝑌) ∈ V)
63, 4, 5syl2anr 597 . . 3 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝑌) ∈ V)
72, 6eqeltrrd 2834 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → 𝑋 ∈ V)
8 fof 6802 . . . . 5 (𝐹:𝑌onto𝑋𝐹:𝑌𝑋)
9 elfm2.l . . . . . 6 𝐿 = (𝑌filGen𝐵)
109elfm2 23443 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
118, 10syl3an3 1165 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
12 fgcl 23373 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
139, 12eqeltrid 2837 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
14133ad2ant2 1134 . . . . . . . . . 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 6077 . . . . . . . . . . . 12 (𝐹𝐴) ⊆ dom 𝐹
18 fofn 6804 . . . . . . . . . . . . 13 (𝐹:𝑌onto𝑋𝐹 Fn 𝑌)
1918fndmd 6651 . . . . . . . . . . . 12 (𝐹:𝑌onto𝑋 → dom 𝐹 = 𝑌)
2017, 19sseqtrid 4033 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → (𝐹𝐴) ⊆ 𝑌)
21203ad2ant3 1135 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐹𝐴) ⊆ 𝑌)
2221ad2antrr 724 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ⊆ 𝑌)
2333ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → Fun 𝐹)
2423ad2antrr 724 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → Fun 𝐹)
259eleq2i 2825 . . . . . . . . . . . . . . 15 (𝑦𝐿𝑦 ∈ (𝑌filGen𝐵))
26 elfg 23366 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
27263ad2ant2 1134 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
2827adantr 481 . . . . . . . . . . . . . . 15 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦 ∈ (𝑌filGen𝐵) ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
2925, 28bitrid 282 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (𝑦𝐿 ↔ (𝑦𝑌 ∧ ∃𝑧𝐵 𝑧𝑦)))
3029simprbda 499 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦𝑌)
31 sseq2 4007 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = 𝑌 → (𝑦 ⊆ dom 𝐹𝑦𝑌))
3231biimpar 478 . . . . . . . . . . . . . . . 16 ((dom 𝐹 = 𝑌𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3319, 32sylan 580 . . . . . . . . . . . . . . 15 ((𝐹:𝑌onto𝑋𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
34333ad2antl3 1187 . . . . . . . . . . . . . 14 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3534adantlr 713 . . . . . . . . . . . . 13 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝑌) → 𝑦 ⊆ dom 𝐹)
3630, 35syldan 591 . . . . . . . . . . . 12 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → 𝑦 ⊆ dom 𝐹)
37 funimass3 7052 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ⊆ dom 𝐹) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
3824, 36, 37syl2anc 584 . . . . . . . . . . 11 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
3938biimpd 228 . . . . . . . . . 10 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ 𝑦𝐿) → ((𝐹𝑦) ⊆ 𝐴𝑦 ⊆ (𝐹𝐴)))
4039impr 455 . . . . . . . . 9 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝑦 ⊆ (𝐹𝐴))
41 filss 23348 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑦𝐿 ∧ (𝐹𝐴) ⊆ 𝑌𝑦 ⊆ (𝐹𝐴))) → (𝐹𝐴) ∈ 𝐿)
4215, 16, 22, 40, 41syl13anc 1372 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → (𝐹𝐴) ∈ 𝐿)
43 foimacnv 6847 . . . . . . . . . . 11 ((𝐹:𝑌onto𝑋𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
4443eqcomd 2738 . . . . . . . . . 10 ((𝐹:𝑌onto𝑋𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
45443ad2antl3 1187 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → 𝐴 = (𝐹 “ (𝐹𝐴)))
4645adantr 481 . . . . . . . 8 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → 𝐴 = (𝐹 “ (𝐹𝐴)))
47 imaeq2 6053 . . . . . . . . 9 (𝑥 = (𝐹𝐴) → (𝐹𝑥) = (𝐹 “ (𝐹𝐴)))
4847rspceeqv 3632 . . . . . . . 8 (((𝐹𝐴) ∈ 𝐿𝐴 = (𝐹 “ (𝐹𝐴))) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
4942, 46, 48syl2anc 584 . . . . . . 7 ((((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) ∧ (𝑦𝐿 ∧ (𝐹𝑦) ⊆ 𝐴)) → ∃𝑥𝐿 𝐴 = (𝐹𝑥))
5049rexlimdvaa 3156 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ 𝐴𝑋) → (∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴 → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
5150expimpd 454 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) → ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
52 simprr 771 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴 = (𝐹𝑥))
53 imassrn 6068 . . . . . . . . 9 (𝐹𝑥) ⊆ ran 𝐹
54 forn 6805 . . . . . . . . . . 11 (𝐹:𝑌onto𝑋 → ran 𝐹 = 𝑋)
55543ad2ant3 1135 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ran 𝐹 = 𝑋)
5655adantr 481 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ran 𝐹 = 𝑋)
5753, 56sseqtrid 4033 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐹𝑥) ⊆ 𝑋)
5852, 57eqsstrd 4019 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → 𝐴𝑋)
59 eqimss2 4040 . . . . . . . . 9 (𝐴 = (𝐹𝑥) → (𝐹𝑥) ⊆ 𝐴)
60 imaeq2 6053 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
6160sseq1d 4012 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ⊆ 𝐴 ↔ (𝐹𝑥) ⊆ 𝐴))
6261rspcev 3612 . . . . . . . . 9 ((𝑥𝐿 ∧ (𝐹𝑥) ⊆ 𝐴) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6359, 62sylan2 593 . . . . . . . 8 ((𝑥𝐿𝐴 = (𝐹𝑥)) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6463adantl 482 . . . . . . 7 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)
6558, 64jca 512 . . . . . 6 (((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) ∧ (𝑥𝐿𝐴 = (𝐹𝑥))) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴))
6665rexlimdvaa 3156 . . . . 5 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (∃𝑥𝐿 𝐴 = (𝐹𝑥) → (𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴)))
6751, 66impbid 211 . . . 4 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → ((𝐴𝑋 ∧ ∃𝑦𝐿 (𝐹𝑦) ⊆ 𝐴) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
6811, 67bitrd 278 . . 3 ((𝑋 ∈ V ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
69683coml 1127 . 2 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋𝑋 ∈ V) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))
707, 69mpd3an3 1462 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  wss 3947  ccnv 5674  dom cdm 5675  ran crn 5676  cima 5678  Fun wfun 6534  wf 6536  ontowfo 6538  cfv 6540  (class class class)co 7405  fBascfbas 20924  filGencfg 20925  Filcfil 23340   FilMap cfm 23428
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-fbas 20933  df-fg 20934  df-fil 23341  df-fm 23433
This theorem is referenced by:  fmid  23455
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