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Theorem filnetlem4 35266
Description: Lemma for filnet 35267. (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 35265 . . . 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 8000 . . . . 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 6760 . . . . 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 23359 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝑋 ∈ 𝐹)
11 xpexg 7737 . . . . . 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 5325 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐻 ∈ V)
149, 13fexd 7229 . . 3 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (2nd β†Ύ 𝐻) ∈ V)
153simpli 485 . . . . . . 7 𝐻 = βˆͺ βˆͺ 𝐷
16 dirdm 18553 . . . . . . . 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 6704 . . . . 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 35260 . . . . . . . . . . . . 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 6704 . . . . . . . . . . . 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 6718 . . . . . . . . . 10 ((tailβ€˜π·):π»βŸΆπ’« dom 𝐷 β†’ (tailβ€˜π·) Fn 𝐻)
28 imaeq2 6056 . . . . . . . . . . . 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 4014 . . . . . . . . . . 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 7089 . . . . . . . . . 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 7996 . . . . . . . . . . . . . . 15 2nd :V–ontoβ†’V
33 fofn 6808 . . . . . . . . . . . . . . 15 (2nd :V–ontoβ†’V β†’ 2nd Fn V)
3432, 33ax-mp 5 . . . . . . . . . . . . . 14 2nd Fn V
35 ssv 4007 . . . . . . . . . . . . . 14 𝐻 βŠ† V
36 fnssres 6674 . . . . . . . . . . . . . 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 6650 . . . . . . . . . . . . 13 ((2nd β†Ύ 𝐻) Fn 𝐻 β†’ Fun (2nd β†Ύ 𝐻))
3937, 38ax-mp 5 . . . . . . . . . . . 12 Fun (2nd β†Ύ 𝐻)
4026ffvelcdmda 7087 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) ∈ 𝒫 dom 𝐷)
4140elpwid 4612 . . . . . . . . . . . . . 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 4024 . . . . . . . . . . . . 13 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) βŠ† 𝐻)
4437fndmi 6654 . . . . . . . . . . . . 13 dom (2nd β†Ύ 𝐻) = 𝐻
4543, 44sseqtrrdi 4034 . . . . . . . . . . . 12 (((𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑑 βŠ† 𝑋) ∧ 𝑓 ∈ 𝐻) β†’ ((tailβ€˜π·)β€˜π‘“) βŠ† dom (2nd β†Ύ 𝐻))
46 funimass4 6957 . . . . . . . . . . . 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 35259 . . . . . . . . . . . . . . . 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 35263 . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . 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 7985 . . . . . . . . . . . . . . . 16 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ (1st β€˜π‘‘) = π‘˜)
7574sseq1d 4014 . . . . . . . . . . . . . . 15 (𝑑 = βŸ¨π‘˜, π‘£βŸ© β†’ ((1st β€˜π‘‘) βŠ† (1st β€˜π‘“) ↔ π‘˜ βŠ† (1st β€˜π‘“)))
7672, 73op2ndd 7986 . . . . . . . . . . . . . . . 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 5840 . . . . . . . . . . . . 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 4639 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘˜ β†’ {𝑛} = {π‘˜})
81 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘˜ β†’ 𝑛 = π‘˜)
8280, 81xpeq12d 5708 . . . . . . . . . . . . . . . 16 (𝑛 = π‘˜ β†’ ({𝑛} Γ— 𝑛) = ({π‘˜} Γ— π‘˜))
8382cbviunv 5044 . . . . . . . . . . . . . . 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 3971 . . . . . . . . . . . . . . . 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 7985 . . . . . . . . . . . . . . 15 (𝑓 = βŸ¨π‘›, π‘šβŸ© β†’ (1st β€˜π‘“) = 𝑛)
9796sseq2d 4015 . . . . . . . . . . . . . 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 5841 . . . . . . . . . . 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 23354 . . . . . . . . . . . . . 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 4500 . . . . . . . . . . . . 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 4005 . . . . . . . . . . . . . . 15 𝑛 βŠ† 𝑛
107 sseq1 4008 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑛 β†’ (π‘˜ βŠ† 𝑛 ↔ 𝑛 βŠ† 𝑛))
108 sseq1 4008 . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . 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 23366 . . . . . . . . . . . 12 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 β‰  βˆ…)
12394snnz 4781 . . . . . . . . . . . . . . . 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 4398 . . . . . . . . . . . . . . 15 (({𝑛} Γ— 𝑛) βŠ† βˆ… ↔ ({𝑛} Γ— 𝑛) = βˆ…)
128 xpeq0 6160 . . . . . . . . . . . . . . 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 4495 . . . . . . . . . . . 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 4011 . . . . . . . . . . . 12 (𝐻 βŠ† βˆ… ↔ βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛) βŠ† βˆ…)
137 ss0b 4398 . . . . . . . . . . . 12 (𝐻 βŠ† βˆ… ↔ 𝐻 = βˆ…)
138 iunss 5049 . . . . . . . . . . . 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 6052 . . . . . . . . . . . 12 dom ( I β†Ύ 𝐻) = 𝐻
1431, 2filnetlem2 35264 . . . . . . . . . . . . . 14 (( I β†Ύ 𝐻) βŠ† 𝐷 ∧ 𝐷 βŠ† (𝐻 Γ— 𝐻))
144143simpli 485 . . . . . . . . . . . . 13 ( I β†Ύ 𝐻) βŠ† 𝐷
145 dmss 5903 . . . . . . . . . . . . 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 4018 . . . . . . . . . . 11 𝐻 βŠ† dom 𝐷
148143simpri 487 . . . . . . . . . . . . 13 𝐷 βŠ† (𝐻 Γ— 𝐻)
149 dmss 5903 . . . . . . . . . . . . 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 5930 . . . . . . . . . . . 12 dom (𝐻 Γ— 𝐻) = 𝐻
152150, 151sseqtri 4019 . . . . . . . . . . 11 dom 𝐷 βŠ† 𝐻
153147, 152eqssi 3999 . . . . . . . . . 10 𝐻 = dom 𝐷
154153tailfb 35262 . . . . . . . . 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 23451 . . . . . . . 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 23352 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ (fBasβ€˜π‘‹))
159 elfg 23375 . . . . . . . 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 23379 . . . . 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 6699 . . . . 5 (𝑓 = (2nd β†Ύ 𝐻) β†’ (𝑓:dom π·βŸΆπ‘‹ ↔ (2nd β†Ύ 𝐻):dom π·βŸΆπ‘‹))
167 oveq2 7417 . . . . . . 7 (𝑓 = (2nd β†Ύ 𝐻) β†’ (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd β†Ύ 𝐻)))
168167fveq1d 6894 . . . . . 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 5904 . . . . . 6 (𝑑 = 𝐷 β†’ dom 𝑑 = dom 𝐷)
174173feq2d 6704 . . . . 5 (𝑑 = 𝐷 β†’ (𝑓:dom π‘‘βŸΆπ‘‹ ↔ 𝑓:dom π·βŸΆπ‘‹))
175 fveq2 6892 . . . . . . . 8 (𝑑 = 𝐷 β†’ (tailβ€˜π‘‘) = (tailβ€˜π·))
176175rneqd 5938 . . . . . . 7 (𝑑 = 𝐷 β†’ ran (tailβ€˜π‘‘) = ran (tailβ€˜π·))
177176fveq2d 6896 . . . . . 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 3949  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βŸ¨cop 4635  βˆͺ cuni 4909  βˆͺ ciun 4998   class class class wbr 5149  {copab 5211   I cid 5574   Γ— cxp 5675  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  DirRelcdir 18547  tailctail 18548  fBascfbas 20932  filGencfg 20933  Filcfil 23349   FilMap cfm 23437
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-dir 18549  df-tail 18550  df-fbas 20941  df-fg 20942  df-fil 23350  df-fm 23442
This theorem is referenced by:  filnet  35267
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