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Theorem filnetlem4 35255
Description: Lemma for filnet 35256. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypotheses
Ref Expression
filnet.h 𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)
filnet.d 𝐷 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}
Assertion
Ref Expression
filnetlem4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
Distinct variable groups:   π‘₯,𝑦   𝑓,𝑑,𝑛,π‘₯,𝑦,𝐹   𝐻,𝑑,𝑓,π‘₯,𝑦   𝐷,𝑑,𝑓   𝑋,𝑑,𝑓,𝑛
Allowed substitution hints:   𝐷(π‘₯,𝑦,𝑛)   𝐻(𝑛)   𝑋(π‘₯,𝑦)
Proof of Theorem filnetlem4
Dummy variables π‘˜ π‘š 𝑑 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filnet.h . . . . 5 𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)
2 filnet.d . . . . 5 𝐷 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}
31, 2filnetlem3 35254 . . . 4 (𝐻 = βˆͺ βˆͺ 𝐷 ∧ (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐻 βŠ† (𝐹 Γ— 𝑋) ∧ 𝐷 ∈ DirRel)))
43simpri 487 . . 3 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐻 βŠ† (𝐹 Γ— 𝑋) ∧ 𝐷 ∈ DirRel))
54simprd 497 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐷 ∈ DirRel)
6 f2ndres 7997 . . . . 5 (2nd β†Ύ (𝐹 Γ— 𝑋)):(𝐹 Γ— 𝑋)βŸΆπ‘‹
74simpld 496 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐻 βŠ† (𝐹 Γ— 𝑋))
8 fssres2 6757 . . . . 5 (((2nd β†Ύ (𝐹 Γ— 𝑋)):(𝐹 Γ— 𝑋)βŸΆπ‘‹ ∧ 𝐻 βŠ† (𝐹 Γ— 𝑋)) β†’ (2nd β†Ύ 𝐻):π»βŸΆπ‘‹)
96, 7, 8sylancr 588 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (2nd β†Ύ 𝐻):π»βŸΆπ‘‹)
10 filtop 23351 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
11 xpexg 7734 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑋 ∈ 𝐹) β†’ (𝐹 Γ— 𝑋) ∈ V)
1210, 11mpdan 686 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐹 Γ— 𝑋) ∈ V)
1312, 7ssexd 5324 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐻 ∈ V)
149, 13fexd 7226 . . 3 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (2nd β†Ύ 𝐻) ∈ V)
153simpli 485 . . . . . . 7 𝐻 = βˆͺ βˆͺ 𝐷
16 dirdm 18550 . . . . . . . 8 (𝐷 ∈ DirRel β†’ dom 𝐷 = βˆͺ βˆͺ 𝐷)
175, 16syl 17 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ dom 𝐷 = βˆͺ βˆͺ 𝐷)
1815, 17eqtr4id 2792 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐻 = dom 𝐷)
1918feq2d 6701 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((2nd β†Ύ 𝐻):π»βŸΆπ‘‹ ↔ (2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹))
209, 19mpbid 231 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹)
21 eqid 2733 . . . . . . . . . . . . . 14 dom 𝐷 = dom 𝐷
2221tailf 35249 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel β†’ (tailβ€˜π·):dom π·βŸΆπ’« dom 𝐷)
235, 22syl 17 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (tailβ€˜π·):dom π·βŸΆπ’« dom 𝐷)
2418feq2d 6701 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((tailβ€˜π·):π»βŸΆπ’« dom 𝐷 ↔ (tailβ€˜π·):dom π·βŸΆπ’« dom 𝐷))
2523, 24mpbird 257 . . . . . . . . . . 11 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (tailβ€˜π·):π»βŸΆπ’« dom 𝐷)
2625adantr 482 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) β†’ (tailβ€˜π·):π»βŸΆπ’« dom 𝐷)
27 ffn 6715 . . . . . . . . . 10 ((tailβ€˜π·):π»βŸΆπ’« dom 𝐷 β†’ (tailβ€˜π·) Fn 𝐻)
28 imaeq2 6054 . . . . . . . . . . . 12 (𝑑 = ((tailβ€˜π·)β€˜π‘“) β†’ ((2nd β†Ύ 𝐻) β€œ 𝑑) = ((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)))
2928sseq1d 4013 . . . . . . . . . . 11 (𝑑 = ((tailβ€˜π·)β€˜π‘“) β†’ (((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑 ↔ ((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑))
3029rexrn 7086 . . . . . . . . . 10 ((tailβ€˜π·) Fn 𝐻 β†’ (βˆƒπ‘‘ ∈ ran (tailβ€˜π·)((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑 ↔ βˆƒπ‘“ ∈ 𝐻 ((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑))
3126, 27, 303syl 18 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) β†’ (βˆƒπ‘‘ ∈ ran (tailβ€˜π·)((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑 ↔ βˆƒπ‘“ ∈ 𝐻 ((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑))
32 fo2nd 7993 . . . . . . . . . . . . . . 15 2nd :V–ontoβ†’V
33 fofn 6805 . . . . . . . . . . . . . . 15 (2nd :V–ontoβ†’V β†’ 2nd Fn V)
3432, 33ax-mp 5 . . . . . . . . . . . . . 14 2nd Fn V
35 ssv 4006 . . . . . . . . . . . . . 14 𝐻 βŠ† V
36 fnssres 6671 . . . . . . . . . . . . . 14 ((2nd Fn V ∧ 𝐻 βŠ† V) β†’ (2nd β†Ύ 𝐻) Fn 𝐻)
3734, 35, 36mp2an 691 . . . . . . . . . . . . 13 (2nd β†Ύ 𝐻) Fn 𝐻
38 fnfun 6647 . . . . . . . . . . . . 13 ((2nd β†Ύ 𝐻) Fn 𝐻 β†’ Fun (2nd β†Ύ 𝐻))
3937, 38ax-mp 5 . . . . . . . . . . . 12 Fun (2nd β†Ύ 𝐻)
4026ffvelcdmda 7084 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) ∈ 𝒫 dom 𝐷)
4140elpwid 4611 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) βŠ† dom 𝐷)
4218ad2antrr 725 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ 𝐻 = dom 𝐷)
4341, 42sseqtrrd 4023 . . . . . . . . . . . . 13 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) βŠ† 𝐻)
4437fndmi 6651 . . . . . . . . . . . . 13 dom (2nd β†Ύ 𝐻) = 𝐻
4543, 44sseqtrrdi 4033 . . . . . . . . . . . 12 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) βŠ† dom (2nd β†Ύ 𝐻))
46 funimass4 6954 . . . . . . . . . . . 12 ((Fun (2nd β†Ύ 𝐻) ∧ ((tailβ€˜π·)β€˜π‘“) βŠ† dom (2nd β†Ύ 𝐻)) β†’ (((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑 ↔ βˆ€π‘‘ ∈ ((tailβ€˜π·)β€˜π‘“)((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑))
4739, 45, 46sylancr 588 . . . . . . . . . . 11 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ (((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑 ↔ βˆ€π‘‘ ∈ ((tailβ€˜π·)β€˜π‘“)((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑))
485ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ 𝐷 ∈ DirRel)
49 simpr 486 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ 𝑓 ∈ 𝐻)
5049, 42eleqtrd 2836 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ 𝑓 ∈ dom 𝐷)
51 vex 3479 . . . . . . . . . . . . . . . . 17 𝑑 ∈ V
5251a1i 11 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ 𝑑 ∈ V)
5321eltail 35248 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑓 ∈ dom 𝐷 ∧ 𝑑 ∈ V) β†’ (𝑑 ∈ ((tailβ€˜π·)β€˜π‘“) ↔ 𝑓𝐷𝑑))
5448, 50, 52, 53syl3anc 1372 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ (𝑑 ∈ ((tailβ€˜π·)β€˜π‘“) ↔ 𝑓𝐷𝑑))
5549biantrurd 534 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ (𝑑 ∈ 𝐻 ↔ (𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻)))
5655anbi1d 631 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) ↔ ((𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻) ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“))))
57 vex 3479 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
581, 2, 57, 51filnetlem1 35252 . . . . . . . . . . . . . . . 16 (𝑓𝐷𝑑 ↔ ((𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻) ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)))
5956, 58bitr4di 289 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) ↔ 𝑓𝐷𝑑))
6054, 59bitr4d 282 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ (𝑑 ∈ ((tailβ€˜π·)β€˜π‘“) ↔ (𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“))))
6160imbi1d 342 . . . . . . . . . . . . 13 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((𝑑 ∈ ((tailβ€˜π·)β€˜π‘“) β†’ ((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑) ↔ ((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) β†’ ((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑)))
62 fvres 6908 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ 𝐻 β†’ ((2nd β†Ύ 𝐻)β€˜π‘‘) = (2nd β€˜π‘‘))
6362eleq1d 2819 . . . . . . . . . . . . . . . 16 (𝑑 ∈ 𝐻 β†’ (((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑 ↔ (2nd β€˜π‘‘) ∈ 𝑑))
6463adantr 482 . . . . . . . . . . . . . . 15 ((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) β†’ (((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑 ↔ (2nd β€˜π‘‘) ∈ 𝑑))
6564pm5.74i 271 . . . . . . . . . . . . . 14 (((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) β†’ ((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑) ↔ ((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) β†’ (2nd β€˜π‘‘) ∈ 𝑑))
66 impexp 452 . . . . . . . . . . . . . 14 (((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ (𝑑 ∈ 𝐻 β†’ ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑)))
6765, 66bitri 275 . . . . . . . . . . . . 13 (((𝑑 ∈ 𝐻 ∧ (1st β€˜π‘‘) βŠ† (1st β€˜π‘“)) β†’ ((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑) ↔ (𝑑 ∈ 𝐻 β†’ ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑)))
6861, 67bitrdi 287 . . . . . . . . . . . 12 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((𝑑 ∈ ((tailβ€˜π·)β€˜π‘“) β†’ ((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑) ↔ (𝑑 ∈ 𝐻 β†’ ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑))))
6968ralbidv2 3174 . . . . . . . . . . 11 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ (βˆ€π‘‘ ∈ ((tailβ€˜π·)β€˜π‘“)((2nd β†Ύ 𝐻)β€˜π‘‘) ∈ 𝑑 ↔ βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑)))
7047, 69bitrd 279 . . . . . . . . . 10 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ (((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑 ↔ βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑)))
7170rexbidva 3177 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) β†’ (βˆƒπ‘“ ∈ 𝐻 ((2nd β†Ύ 𝐻) β€œ ((tailβ€˜π·)β€˜π‘“)) βŠ† 𝑑 ↔ βˆƒπ‘“ ∈ 𝐻 βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑)))
72 vex 3479 . . . . . . . . . . . . . . . . 17 π‘˜ ∈ V
73 vex 3479 . . . . . . . . . . . . . . . . 17 𝑣 ∈ V
7472, 73op1std 7982 . . . . . . . . . . . . . . . 16 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ (1st β€˜π‘‘) = π‘˜)
7574sseq1d 4013 . . . . . . . . . . . . . . 15 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) ↔ π‘˜ βŠ† (1st β€˜π‘“)))
7672, 73op2ndd 7983 . . . . . . . . . . . . . . . 16 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ (2nd β€˜π‘‘) = 𝑣)
7776eleq1d 2819 . . . . . . . . . . . . . . 15 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ ((2nd β€˜π‘‘) ∈ 𝑑 ↔ 𝑣 ∈ 𝑑))
7875, 77imbi12d 345 . . . . . . . . . . . . . 14 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ (((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ (π‘˜ βŠ† (1st β€˜π‘“) β†’ 𝑣 ∈ 𝑑)))
7978raliunxp 5838 . . . . . . . . . . . . 13 (βˆ€π‘‘ ∈ βˆͺ π‘˜ ∈ 𝐹 ({π‘˜} Γ— π‘˜)((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ βˆ€π‘˜ ∈ 𝐹 βˆ€π‘£ ∈ π‘˜ (π‘˜ βŠ† (1st β€˜π‘“) β†’ 𝑣 ∈ 𝑑))
80 sneq 4638 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘˜ β†’ {𝑛} = {π‘˜})
81 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘˜ β†’ 𝑛 = π‘˜)
8280, 81xpeq12d 5707 . . . . . . . . . . . . . . . 16 (𝑛 = π‘˜ β†’ ({𝑛} Γ— 𝑛) = ({π‘˜} Γ— π‘˜))
8382cbviunv 5043 . . . . . . . . . . . . . . 15 βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛) = βˆͺ π‘˜ ∈ 𝐹 ({π‘˜} Γ— π‘˜)
841, 83eqtri 2761 . . . . . . . . . . . . . 14 𝐻 = βˆͺ π‘˜ ∈ 𝐹 ({π‘˜} Γ— π‘˜)
8584raleqi 3324 . . . . . . . . . . . . 13 (βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ βˆ€π‘‘ ∈ βˆͺ π‘˜ ∈ 𝐹 ({π‘˜} Γ— π‘˜)((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑))
86 dfss3 3970 . . . . . . . . . . . . . . . 16 (π‘˜ βŠ† 𝑑 ↔ βˆ€π‘£ ∈ π‘˜ 𝑣 ∈ 𝑑)
8786imbi2i 336 . . . . . . . . . . . . . . 15 ((π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ (π‘˜ βŠ† (1st β€˜π‘“) β†’ βˆ€π‘£ ∈ π‘˜ 𝑣 ∈ 𝑑))
88 r19.21v 3180 . . . . . . . . . . . . . . 15 (βˆ€π‘£ ∈ π‘˜ (π‘˜ βŠ† (1st β€˜π‘“) β†’ 𝑣 ∈ 𝑑) ↔ (π‘˜ βŠ† (1st β€˜π‘“) β†’ βˆ€π‘£ ∈ π‘˜ 𝑣 ∈ 𝑑))
8987, 88bitr4i 278 . . . . . . . . . . . . . 14 ((π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ βˆ€π‘£ ∈ π‘˜ (π‘˜ βŠ† (1st β€˜π‘“) β†’ 𝑣 ∈ 𝑑))
9089ralbii 3094 . . . . . . . . . . . . 13 (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ βˆ€π‘˜ ∈ 𝐹 βˆ€π‘£ ∈ π‘˜ (π‘˜ βŠ† (1st β€˜π‘“) β†’ 𝑣 ∈ 𝑑))
9179, 85, 903bitr4i 303 . . . . . . . . . . . 12 (βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑))
9291rexbii 3095 . . . . . . . . . . 11 (βˆƒπ‘“ ∈ 𝐻 βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ βˆƒπ‘“ ∈ 𝐻 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑))
931rexeqi 3325 . . . . . . . . . . 11 (βˆƒπ‘“ ∈ 𝐻 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ βˆƒπ‘“ ∈ βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑))
94 vex 3479 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
95 vex 3479 . . . . . . . . . . . . . . . 16 π‘š ∈ V
9694, 95op1std 7982 . . . . . . . . . . . . . . 15 (𝑓 = βŸ¨π‘›, π‘šβŸ© β†’ (1st β€˜π‘“) = 𝑛)
9796sseq2d 4014 . . . . . . . . . . . . . 14 (𝑓 = βŸ¨π‘›, π‘šβŸ© β†’ (π‘˜ βŠ† (1st β€˜π‘“) ↔ π‘˜ βŠ† 𝑛))
9897imbi1d 342 . . . . . . . . . . . . 13 (𝑓 = βŸ¨π‘›, π‘šβŸ© β†’ ((π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑)))
9998ralbidv 3178 . . . . . . . . . . . 12 (𝑓 = βŸ¨π‘›, π‘šβŸ© β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑)))
10099rexiunxp 5839 . . . . . . . . . . 11 (βˆƒπ‘“ ∈ βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† (1st β€˜π‘“) β†’ π‘˜ βŠ† 𝑑) ↔ βˆƒπ‘› ∈ 𝐹 βˆƒπ‘š ∈ 𝑛 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑))
10192, 93, 1003bitri 297 . . . . . . . . . 10 (βˆƒπ‘“ ∈ 𝐻 βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ βˆƒπ‘› ∈ 𝐹 βˆƒπ‘š ∈ 𝑛 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑))
102 fileln0 23346 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑛 ∈ 𝐹) β†’ 𝑛 β‰  βˆ…)
103102adantlr 714 . . . . . . . . . . . . 13 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑛 ∈ 𝐹) β†’ 𝑛 β‰  βˆ…)
104 r19.9rzv 4499 . . . . . . . . . . . . 13 (𝑛 β‰  βˆ… β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) ↔ βˆƒπ‘š ∈ 𝑛 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑)))
105103, 104syl 17 . . . . . . . . . . . 12 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑛 ∈ 𝐹) β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) ↔ βˆƒπ‘š ∈ 𝑛 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑)))
106 ssid 4004 . . . . . . . . . . . . . . 15 𝑛 βŠ† 𝑛
107 sseq1 4007 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑛 β†’ (π‘˜ βŠ† 𝑛 ↔ 𝑛 βŠ† 𝑛))
108 sseq1 4007 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑛 β†’ (π‘˜ βŠ† 𝑑 ↔ 𝑛 βŠ† 𝑑))
109107, 108imbi12d 345 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝑛 β†’ ((π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) ↔ (𝑛 βŠ† 𝑛 β†’ 𝑛 βŠ† 𝑑)))
110109rspcv 3609 . . . . . . . . . . . . . . 15 (𝑛 ∈ 𝐹 β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) β†’ (𝑛 βŠ† 𝑛 β†’ 𝑛 βŠ† 𝑑)))
111106, 110mpii 46 . . . . . . . . . . . . . 14 (𝑛 ∈ 𝐹 β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) β†’ 𝑛 βŠ† 𝑑))
112111adantl 483 . . . . . . . . . . . . 13 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑛 ∈ 𝐹) β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) β†’ 𝑛 βŠ† 𝑑))
113 sstr2 3989 . . . . . . . . . . . . . . 15 (π‘˜ βŠ† 𝑛 β†’ (𝑛 βŠ† 𝑑 β†’ π‘˜ βŠ† 𝑑))
114113com12 32 . . . . . . . . . . . . . 14 (𝑛 βŠ† 𝑑 β†’ (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑))
115114ralrimivw 3151 . . . . . . . . . . . . 13 (𝑛 βŠ† 𝑑 β†’ βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑))
116112, 115impbid1 224 . . . . . . . . . . . 12 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑛 ∈ 𝐹) β†’ (βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) ↔ 𝑛 βŠ† 𝑑))
117105, 116bitr3d 281 . . . . . . . . . . 11 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑛 ∈ 𝐹) β†’ (βˆƒπ‘š ∈ 𝑛 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) ↔ 𝑛 βŠ† 𝑑))
118117rexbidva 3177 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) β†’ (βˆƒπ‘› ∈ 𝐹 βˆƒπ‘š ∈ 𝑛 βˆ€π‘˜ ∈ 𝐹 (π‘˜ βŠ† 𝑛 β†’ π‘˜ βŠ† 𝑑) ↔ βˆƒπ‘› ∈ 𝐹 𝑛 βŠ† 𝑑))
119101, 118bitrid 283 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) β†’ (βˆƒπ‘“ ∈ 𝐻 βˆ€π‘‘ ∈ 𝐻 ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) β†’ (2nd β€˜π‘‘) ∈ 𝑑) ↔ βˆƒπ‘› ∈ 𝐹 𝑛 βŠ† 𝑑))
12031, 71, 1193bitrd 305 . . . . . . . 8 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) β†’ (βˆƒπ‘‘ ∈ ran (tailβ€˜π·)((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑 ↔ βˆƒπ‘› ∈ 𝐹 𝑛 βŠ† 𝑑))
121120pm5.32da 580 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝑑 βŠ† 𝑋 ∧ βˆƒπ‘‘ ∈ ran (tailβ€˜π·)((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑) ↔ (𝑑 βŠ† 𝑋 ∧ βˆƒπ‘› ∈ 𝐹 𝑛 βŠ† 𝑑)))
122 filn0 23358 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
12394snnz 4780 . . . . . . . . . . . . . . . 16 {𝑛} β‰  βˆ…
124102, 123jctil 521 . . . . . . . . . . . . . . 15 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑛 ∈ 𝐹) β†’ ({𝑛} β‰  βˆ… ∧ 𝑛 β‰  βˆ…))
125 neanior 3036 . . . . . . . . . . . . . . 15 (({𝑛} β‰  βˆ… ∧ 𝑛 β‰  βˆ…) ↔ Β¬ ({𝑛} = βˆ… ∨ 𝑛 = βˆ…))
126124, 125sylib 217 . . . . . . . . . . . . . 14 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑛 ∈ 𝐹) β†’ Β¬ ({𝑛} = βˆ… ∨ 𝑛 = βˆ…))
127 ss0b 4397 . . . . . . . . . . . . . . 15 (({𝑛} Γ— 𝑛) βŠ† βˆ… ↔ ({𝑛} Γ— 𝑛) = βˆ…)
128 xpeq0 6157 . . . . . . . . . . . . . . 15 (({𝑛} Γ— 𝑛) = βˆ… ↔ ({𝑛} = βˆ… ∨ 𝑛 = βˆ…))
129127, 128bitri 275 . . . . . . . . . . . . . 14 (({𝑛} Γ— 𝑛) βŠ† βˆ… ↔ ({𝑛} = βˆ… ∨ 𝑛 = βˆ…))
130126, 129sylnibr 329 . . . . . . . . . . . . 13 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑛 ∈ 𝐹) β†’ Β¬ ({𝑛} Γ— 𝑛) βŠ† βˆ…)
131130ralrimiva 3147 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆ€π‘› ∈ 𝐹 Β¬ ({𝑛} Γ— 𝑛) βŠ† βˆ…)
132 r19.2z 4494 . . . . . . . . . . . 12 ((𝐹 β‰  βˆ… ∧ βˆ€π‘› ∈ 𝐹 Β¬ ({𝑛} Γ— 𝑛) βŠ† βˆ…) β†’ βˆƒπ‘› ∈ 𝐹 Β¬ ({𝑛} Γ— 𝑛) βŠ† βˆ…)
133122, 131, 132syl2anc 585 . . . . . . . . . . 11 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘› ∈ 𝐹 Β¬ ({𝑛} Γ— 𝑛) βŠ† βˆ…)
134 rexnal 3101 . . . . . . . . . . 11 (βˆƒπ‘› ∈ 𝐹 Β¬ ({𝑛} Γ— 𝑛) βŠ† βˆ… ↔ Β¬ βˆ€π‘› ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
135133, 134sylib 217 . . . . . . . . . 10 (𝐹 ∈ (Filβ€˜π‘‹) β†’ Β¬ βˆ€π‘› ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
1361sseq1i 4010 . . . . . . . . . . . 12 (𝐻 βŠ† βˆ… ↔ βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
137 ss0b 4397 . . . . . . . . . . . 12 (𝐻 βŠ† βˆ… ↔ 𝐻 = βˆ…)
138 iunss 5048 . . . . . . . . . . . 12 (βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ… ↔ βˆ€π‘› ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
139136, 137, 1383bitr3i 301 . . . . . . . . . . 11 (𝐻 = βˆ… ↔ βˆ€π‘› ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
140139necon3abii 2988 . . . . . . . . . 10 (𝐻 β‰  βˆ… ↔ Β¬ βˆ€π‘› ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
141135, 140sylibr 233 . . . . . . . . 9 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐻 β‰  βˆ…)
142 dmresi 6050 . . . . . . . . . . . 12 dom ( I β†Ύ 𝐻) = 𝐻
1431, 2filnetlem2 35253 . . . . . . . . . . . . . 14 (( I β†Ύ 𝐻) βŠ† 𝐷 ∧ 𝐷 βŠ† (𝐻 Γ— 𝐻))
144143simpli 485 . . . . . . . . . . . . 13 ( I β†Ύ 𝐻) βŠ† 𝐷
145 dmss 5901 . . . . . . . . . . . . 13 (( I β†Ύ 𝐻) βŠ† 𝐷 β†’ dom ( I β†Ύ 𝐻) βŠ† dom 𝐷)
146144, 145ax-mp 5 . . . . . . . . . . . 12 dom ( I β†Ύ 𝐻) βŠ† dom 𝐷
147142, 146eqsstrri 4017 . . . . . . . . . . 11 𝐻 βŠ† dom 𝐷
148143simpri 487 . . . . . . . . . . . . 13 𝐷 βŠ† (𝐻 Γ— 𝐻)
149 dmss 5901 . . . . . . . . . . . . 13 (𝐷 βŠ† (𝐻 Γ— 𝐻) β†’ dom 𝐷 βŠ† dom (𝐻 Γ— 𝐻))
150148, 149ax-mp 5 . . . . . . . . . . . 12 dom 𝐷 βŠ† dom (𝐻 Γ— 𝐻)
151 dmxpid 5928 . . . . . . . . . . . 12 dom (𝐻 Γ— 𝐻) = 𝐻
152150, 151sseqtri 4018 . . . . . . . . . . 11 dom 𝐷 βŠ† 𝐻
153147, 152eqssi 3998 . . . . . . . . . 10 𝐻 = dom 𝐷
154153tailfb 35251 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ 𝐻 β‰  βˆ…) β†’ ran (tailβ€˜π·) ∈ (fBasβ€˜π»))
1555, 141, 154syl2anc 585 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ran (tailβ€˜π·) ∈ (fBasβ€˜π»))
156 elfm 23443 . . . . . . . 8 ((𝑋 ∈ 𝐹 ∧ ran (tailβ€˜π·) ∈ (fBasβ€˜π») ∧ (2nd β†Ύ 𝐻):π»βŸΆπ‘‹) β†’ (𝑑 ∈ ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)) ↔ (𝑑 βŠ† 𝑋 ∧ βˆƒπ‘‘ ∈ ran (tailβ€˜π·)((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑)))
15710, 155, 9, 156syl3anc 1372 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝑑 ∈ ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)) ↔ (𝑑 βŠ† 𝑋 ∧ βˆƒπ‘‘ ∈ ran (tailβ€˜π·)((2nd β†Ύ 𝐻) β€œ 𝑑) βŠ† 𝑑)))
158 filfbas 23344 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
159 elfg 23367 . . . . . . . 8 (𝐹 ∈ (fBasβ€˜π‘‹) β†’ (𝑑 ∈ (𝑋filGen𝐹) ↔ (𝑑 βŠ† 𝑋 ∧ βˆƒπ‘› ∈ 𝐹 𝑛 βŠ† 𝑑)))
160158, 159syl 17 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝑑 ∈ (𝑋filGen𝐹) ↔ (𝑑 βŠ† 𝑋 ∧ βˆƒπ‘› ∈ 𝐹 𝑛 βŠ† 𝑑)))
161121, 157, 1603bitr4d 311 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝑑 ∈ ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)) ↔ 𝑑 ∈ (𝑋filGen𝐹)))
162161eqrdv 2731 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)) = (𝑋filGen𝐹))
163 fgfil 23371 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝑋filGen𝐹) = 𝐹)
164162, 163eqtr2d 2774 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 = ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)))
16520, 164jca 513 . . 3 (𝐹 ∈ (Filβ€˜π‘‹) β†’ ((2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·))))
166 feq1 6696 . . . . 5 (𝑓 = (2nd β†Ύ 𝐻) β†’ (𝑓:dom π·βŸΆπ‘‹ ↔ (2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹))
167 oveq2 7414 . . . . . . 7 (𝑓 = (2nd β†Ύ 𝐻) β†’ (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd β†Ύ 𝐻)))
168167fveq1d 6891 . . . . . 6 (𝑓 = (2nd β†Ύ 𝐻) β†’ ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)) = ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)))
169168eqeq2d 2744 . . . . 5 (𝑓 = (2nd β†Ύ 𝐻) β†’ (𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)) ↔ 𝐹 = ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·))))
170166, 169anbi12d 632 . . . 4 (𝑓 = (2nd β†Ύ 𝐻) β†’ ((𝑓:dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·))) ↔ ((2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·)))))
171170spcegv 3588 . . 3 ((2nd β†Ύ 𝐻) ∈ V β†’ (((2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap (2nd β†Ύ 𝐻))β€˜ran (tailβ€˜π·))) β†’ βˆƒπ‘“(𝑓:dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)))))
17214, 165, 171sylc 65 . 2 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘“(𝑓:dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·))))
173 dmeq 5902 . . . . . 6 (𝑑 = 𝐷 β†’ dom 𝑑 = dom 𝐷)
174173feq2d 6701 . . . . 5 (𝑑 = 𝐷 β†’ (𝑓:dom π‘‘βŸΆπ‘‹ ↔ 𝑓:dom π·βŸΆπ‘‹))
175 fveq2 6889 . . . . . . . 8 (𝑑 = 𝐷 β†’ (tailβ€˜π‘‘) = (tailβ€˜π·))
176175rneqd 5936 . . . . . . 7 (𝑑 = 𝐷 β†’ ran (tailβ€˜π‘‘) = ran (tailβ€˜π·))
177176fveq2d 6893 . . . . . 6 (𝑑 = 𝐷 β†’ ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘)) = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)))
178177eqeq2d 2744 . . . . 5 (𝑑 = 𝐷 β†’ (𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘)) ↔ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·))))
179174, 178anbi12d 632 . . . 4 (𝑑 = 𝐷 β†’ ((𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))) ↔ (𝑓:dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)))))
180179exbidv 1925 . . 3 (𝑑 = 𝐷 β†’ (βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))) ↔ βˆƒπ‘“(𝑓:dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)))))
181180rspcev 3613 . 2 ((𝐷 ∈ DirRel ∧ βˆƒπ‘“(𝑓:dom π·βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π·)))) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
1825, 172, 181syl2anc 585 1 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908  βˆͺ ciun 4997   class class class wbr 5148  {copab 5210   I cid 5573   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6535   Fn wfn 6536  βŸΆwf 6537  β€“ontoβ†’wfo 6539  β€˜cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971  DirRelcdir 18544  tailctail 18545  fBascfbas 20925  filGencfg 20926  Filcfil 23341   FilMap cfm 23429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-dir 18546  df-tail 18547  df-fbas 20934  df-fg 20935  df-fil 23342  df-fm 23434
This theorem is referenced by:  filnet  35256
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