Metamath Proof Explorer

Theorem elfm3

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)

Ref Expression
Hypothesis elfm2.l ⊢ 𝐿 = ( 𝑌 filGen 𝐵 )
Assertion elfm3 ( ( 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝐴 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 elfm2.l ⊢ 𝐿 = ( 𝑌 filGen 𝐵 )
2 foima ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → ( 𝐹 “ 𝑌 ) = 𝑋 )
3 2 adantl ⊢ ( ( 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝐹 “ 𝑌 ) = 𝑋 )
4 fofun ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → Fun 𝐹 )
5 elfvdm ⊢ ( 𝐵 ∈ ( fBas ‘ 𝑌 ) → 𝑌 ∈ dom fBas )
6 funimaexg ⊢ ( ( Fun 𝐹 ∧ 𝑌 ∈ dom fBas ) → ( 𝐹 “ 𝑌 ) ∈ V )
7 4 5 6 syl2anr ⊢ ( ( 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝐹 “ 𝑌 ) ∈ V )
8 3 7 eqeltrrd ⊢ ( ( 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → 𝑋 ∈ V )
9 fof ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → 𝐹 : 𝑌 ⟶ 𝑋 )
10 1 elfm2 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐴 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐵 ) ↔ ( 𝐴 ⊆ 𝑋 ∧ ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) )
11 9 10 syl3an3 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝐴 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐵 ) ↔ ( 𝐴 ⊆ 𝑋 ∧ ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) )
12 fgcl ⊢ ( 𝐵 ∈ ( fBas ‘ 𝑌 ) → ( 𝑌 filGen 𝐵 ) ∈ ( Fil ‘ 𝑌 ) )
13 1 12 eqeltrid ⊢ ( 𝐵 ∈ ( fBas ‘ 𝑌 ) → 𝐿 ∈ ( Fil ‘ 𝑌 ) )
14 13 3ad2ant2 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → 𝐿 ∈ ( Fil ‘ 𝑌 ) )
15 14 ad2antrr ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → 𝐿 ∈ ( Fil ‘ 𝑌 ) )
16 simprl ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → 𝑦 ∈ 𝐿 )
17 cnvimass ⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ dom 𝐹
18 fofn ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → 𝐹 Fn 𝑌 )
19 18 fndmd ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → dom 𝐹 = 𝑌 )
20 17 19 sseqtrid ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → ( ◡ 𝐹 “ 𝐴 ) ⊆ 𝑌 )
21 20 3ad2ant3 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( ◡ 𝐹 “ 𝐴 ) ⊆ 𝑌 )
22 21 ad2antrr ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → ( ◡ 𝐹 “ 𝐴 ) ⊆ 𝑌 )
23 4 3ad2ant3 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → Fun 𝐹 )
24 23 ad2antrr ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐿 ) → Fun 𝐹 )
25 1 eleq2i ⊢ ( 𝑦 ∈ 𝐿 ↔ 𝑦 ∈ ( 𝑌 filGen 𝐵 ) )
26 elfg ⊢ ( 𝐵 ∈ ( fBas ‘ 𝑌 ) → ( 𝑦 ∈ ( 𝑌 filGen 𝐵 ) ↔ ( 𝑦 ⊆ 𝑌 ∧ ∃ 𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦 ) ) )
27 26 3ad2ant2 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝑦 ∈ ( 𝑌 filGen 𝐵 ) ↔ ( 𝑦 ⊆ 𝑌 ∧ ∃ 𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦 ) ) )
28 27 adantr ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( 𝑦 ∈ ( 𝑌 filGen 𝐵 ) ↔ ( 𝑦 ⊆ 𝑌 ∧ ∃ 𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦 ) ) )
29 25 28 syl5bb ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( 𝑦 ∈ 𝐿 ↔ ( 𝑦 ⊆ 𝑌 ∧ ∃ 𝑧 ∈ 𝐵 𝑧 ⊆ 𝑦 ) ) )
30 29 simprbda ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐿 ) → 𝑦 ⊆ 𝑌 )
31 sseq2 ⊢ ( dom 𝐹 = 𝑌 → ( 𝑦 ⊆ dom 𝐹 ↔ 𝑦 ⊆ 𝑌 ) )
32 31 biimpar ⊢ ( ( dom 𝐹 = 𝑌 ∧ 𝑦 ⊆ 𝑌 ) → 𝑦 ⊆ dom 𝐹 )
33 19 32 sylan ⊢ ( ( 𝐹 : 𝑌 –onto→ 𝑋 ∧ 𝑦 ⊆ 𝑌 ) → 𝑦 ⊆ dom 𝐹 )
34 33 3ad2antl3 ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝑦 ⊆ 𝑌 ) → 𝑦 ⊆ dom 𝐹 )
35 34 adantlr ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑦 ⊆ 𝑌 ) → 𝑦 ⊆ dom 𝐹 )
36 30 35 syldan ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐿 ) → 𝑦 ⊆ dom 𝐹 )
37 funimass3 ⊢ ( ( Fun 𝐹 ∧ 𝑦 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ↔ 𝑦 ⊆ ( ◡ 𝐹 “ 𝐴 ) ) )
38 24 36 37 syl2anc ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐿 ) → ( ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ↔ 𝑦 ⊆ ( ◡ 𝐹 “ 𝐴 ) ) )
39 38 biimpd ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑦 ∈ 𝐿 ) → ( ( 𝐹 “ 𝑦 ) ⊆ 𝐴 → 𝑦 ⊆ ( ◡ 𝐹 “ 𝐴 ) ) )
40 39 impr ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → 𝑦 ⊆ ( ◡ 𝐹 “ 𝐴 ) )
41 filss ⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( ◡ 𝐹 “ 𝐴 ) ⊆ 𝑌 ∧ 𝑦 ⊆ ( ◡ 𝐹 “ 𝐴 ) ) ) → ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐿 )
42 15 16 22 40 41 syl13anc ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐿 )
43 foimacnv ⊢ ( ( 𝐹 : 𝑌 –onto→ 𝑋 ∧ 𝐴 ⊆ 𝑋 ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝐴 ) ) = 𝐴 )
44 43 eqcomd ⊢ ( ( 𝐹 : 𝑌 –onto→ 𝑋 ∧ 𝐴 ⊆ 𝑋 ) → 𝐴 = ( 𝐹 “ ( ◡ 𝐹 “ 𝐴 ) ) )
45 44 3ad2antl3 ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → 𝐴 = ( 𝐹 “ ( ◡ 𝐹 “ 𝐴 ) ) )
46 45 adantr ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → 𝐴 = ( 𝐹 “ ( ◡ 𝐹 “ 𝐴 ) ) )
47 imaeq2 ⊢ ( 𝑥 = ( ◡ 𝐹 “ 𝐴 ) → ( 𝐹 “ 𝑥 ) = ( 𝐹 “ ( ◡ 𝐹 “ 𝐴 ) ) )
48 47 rspceeqv ⊢ ( ( ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ ( ◡ 𝐹 “ 𝐴 ) ) ) → ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) )
49 42 46 48 syl2anc ⊢ ( ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ ( 𝑦 ∈ 𝐿 ∧ ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) → ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) )
50 49 rexlimdvaa ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 → ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )
51 50 expimpd ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( ( 𝐴 ⊆ 𝑋 ∧ ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) → ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )
52 simprr ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) ) → 𝐴 = ( 𝐹 “ 𝑥 ) )
53 imassrn ⊢ ( 𝐹 “ 𝑥 ) ⊆ ran 𝐹
54 forn ⊢ ( 𝐹 : 𝑌 –onto→ 𝑋 → ran 𝐹 = 𝑋 )
55 54 3ad2ant3 ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ran 𝐹 = 𝑋 )
56 55 adantr ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) ) → ran 𝐹 = 𝑋 )
57 53 56 sseqtrid ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) ) → ( 𝐹 “ 𝑥 ) ⊆ 𝑋 )
58 52 57 eqsstrd ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) ) → 𝐴 ⊆ 𝑋 )
59 eqimss2 ⊢ ( 𝐴 = ( 𝐹 “ 𝑥 ) → ( 𝐹 “ 𝑥 ) ⊆ 𝐴 )
60 imaeq2 ⊢ ( 𝑦 = 𝑥 → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ 𝑥 ) )
61 60 sseq1d ⊢ ( 𝑦 = 𝑥 → ( ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ↔ ( 𝐹 “ 𝑥 ) ⊆ 𝐴 ) )
62 61 rspcev ⊢ ( ( 𝑥 ∈ 𝐿 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝐴 ) → ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 )
63 59 62 sylan2 ⊢ ( ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) → ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 )
64 63 adantl ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) ) → ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 )
65 58 64 jca ⊢ ( ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) ∧ ( 𝑥 ∈ 𝐿 ∧ 𝐴 = ( 𝐹 “ 𝑥 ) ) ) → ( 𝐴 ⊆ 𝑋 ∧ ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) )
66 65 rexlimdvaa ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) → ( 𝐴 ⊆ 𝑋 ∧ ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ) )
67 51 66 impbid ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( ( 𝐴 ⊆ 𝑋 ∧ ∃ 𝑦 ∈ 𝐿 ( 𝐹 “ 𝑦 ) ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )
68 11 67 bitrd ⊢ ( ( 𝑋 ∈ V ∧ 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝐴 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )
69 68 3coml ⊢ ( ( 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ∧ 𝑋 ∈ V ) → ( 𝐴 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )
70 8 69 mpd3an3 ⊢ ( ( 𝐵 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 –onto→ 𝑋 ) → ( 𝐴 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐿 𝐴 = ( 𝐹 “ 𝑥 ) ) )