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Theorem tailfb 35310
Description: The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypothesis
Ref Expression
tailfb.1 𝑋 = dom 𝐷
Assertion
Ref Expression
tailfb ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ran (tailβ€˜π·) ∈ (fBasβ€˜π‘‹))

Proof of Theorem tailfb
Dummy variables 𝑣 𝑒 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tailfb.1 . . . . 5 𝑋 = dom 𝐷
21tailf 35308 . . . 4 (𝐷 ∈ DirRel β†’ (tailβ€˜π·):π‘‹βŸΆπ’« 𝑋)
32frnd 6726 . . 3 (𝐷 ∈ DirRel β†’ ran (tailβ€˜π·) βŠ† 𝒫 𝑋)
43adantr 482 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ran (tailβ€˜π·) βŠ† 𝒫 𝑋)
5 n0 4347 . . . . 5 (𝑋 β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ 𝑋)
6 ffn 6718 . . . . . . . 8 ((tailβ€˜π·):π‘‹βŸΆπ’« 𝑋 β†’ (tailβ€˜π·) Fn 𝑋)
7 fnfvelrn 7083 . . . . . . . . 9 (((tailβ€˜π·) Fn 𝑋 ∧ π‘₯ ∈ 𝑋) β†’ ((tailβ€˜π·)β€˜π‘₯) ∈ ran (tailβ€˜π·))
87ex 414 . . . . . . . 8 ((tailβ€˜π·) Fn 𝑋 β†’ (π‘₯ ∈ 𝑋 β†’ ((tailβ€˜π·)β€˜π‘₯) ∈ ran (tailβ€˜π·)))
92, 6, 83syl 18 . . . . . . 7 (𝐷 ∈ DirRel β†’ (π‘₯ ∈ 𝑋 β†’ ((tailβ€˜π·)β€˜π‘₯) ∈ ran (tailβ€˜π·)))
10 ne0i 4335 . . . . . . 7 (((tailβ€˜π·)β€˜π‘₯) ∈ ran (tailβ€˜π·) β†’ ran (tailβ€˜π·) β‰  βˆ…)
119, 10syl6 35 . . . . . 6 (𝐷 ∈ DirRel β†’ (π‘₯ ∈ 𝑋 β†’ ran (tailβ€˜π·) β‰  βˆ…))
1211exlimdv 1937 . . . . 5 (𝐷 ∈ DirRel β†’ (βˆƒπ‘₯ π‘₯ ∈ 𝑋 β†’ ran (tailβ€˜π·) β‰  βˆ…))
135, 12biimtrid 241 . . . 4 (𝐷 ∈ DirRel β†’ (𝑋 β‰  βˆ… β†’ ran (tailβ€˜π·) β‰  βˆ…))
1413imp 408 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ran (tailβ€˜π·) β‰  βˆ…)
151tailini 35309 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ ((tailβ€˜π·)β€˜π‘₯))
16 n0i 4334 . . . . . . . 8 (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘₯) β†’ Β¬ ((tailβ€˜π·)β€˜π‘₯) = βˆ…)
1715, 16syl 17 . . . . . . 7 ((𝐷 ∈ DirRel ∧ π‘₯ ∈ 𝑋) β†’ Β¬ ((tailβ€˜π·)β€˜π‘₯) = βˆ…)
1817nrexdv 3150 . . . . . 6 (𝐷 ∈ DirRel β†’ Β¬ βˆƒπ‘₯ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘₯) = βˆ…)
1918adantr 482 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ Β¬ βˆƒπ‘₯ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘₯) = βˆ…)
20 fvelrnb 6953 . . . . . . 7 ((tailβ€˜π·) Fn 𝑋 β†’ (βˆ… ∈ ran (tailβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘₯) = βˆ…))
212, 6, 203syl 18 . . . . . 6 (𝐷 ∈ DirRel β†’ (βˆ… ∈ ran (tailβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘₯) = βˆ…))
2221adantr 482 . . . . 5 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ (βˆ… ∈ ran (tailβ€˜π·) ↔ βˆƒπ‘₯ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘₯) = βˆ…))
2319, 22mtbird 325 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ Β¬ βˆ… ∈ ran (tailβ€˜π·))
24 df-nel 3048 . . . 4 (βˆ… βˆ‰ ran (tailβ€˜π·) ↔ Β¬ βˆ… ∈ ran (tailβ€˜π·))
2523, 24sylibr 233 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ βˆ… βˆ‰ ran (tailβ€˜π·))
26 fvelrnb 6953 . . . . . . . 8 ((tailβ€˜π·) Fn 𝑋 β†’ (π‘₯ ∈ ran (tailβ€˜π·) ↔ βˆƒπ‘’ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘’) = π‘₯))
27 fvelrnb 6953 . . . . . . . 8 ((tailβ€˜π·) Fn 𝑋 β†’ (𝑦 ∈ ran (tailβ€˜π·) ↔ βˆƒπ‘£ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘£) = 𝑦))
2826, 27anbi12d 632 . . . . . . 7 ((tailβ€˜π·) Fn 𝑋 β†’ ((π‘₯ ∈ ran (tailβ€˜π·) ∧ 𝑦 ∈ ran (tailβ€˜π·)) ↔ (βˆƒπ‘’ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ βˆƒπ‘£ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘£) = 𝑦)))
292, 6, 283syl 18 . . . . . 6 (𝐷 ∈ DirRel β†’ ((π‘₯ ∈ ran (tailβ€˜π·) ∧ 𝑦 ∈ ran (tailβ€˜π·)) ↔ (βˆƒπ‘’ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ βˆƒπ‘£ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘£) = 𝑦)))
30 reeanv 3227 . . . . . . 7 (βˆƒπ‘’ ∈ 𝑋 βˆƒπ‘£ ∈ 𝑋 (((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ ((tailβ€˜π·)β€˜π‘£) = 𝑦) ↔ (βˆƒπ‘’ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ βˆƒπ‘£ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘£) = 𝑦))
311dirge 18556 . . . . . . . . . . 11 ((𝐷 ∈ DirRel ∧ 𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ βˆƒπ‘€ ∈ 𝑋 (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))
32313expb 1121 . . . . . . . . . 10 ((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ βˆƒπ‘€ ∈ 𝑋 (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))
332, 6syl 17 . . . . . . . . . . . . 13 (𝐷 ∈ DirRel β†’ (tailβ€˜π·) Fn 𝑋)
34 fnfvelrn 7083 . . . . . . . . . . . . 13 (((tailβ€˜π·) Fn 𝑋 ∧ 𝑀 ∈ 𝑋) β†’ ((tailβ€˜π·)β€˜π‘€) ∈ ran (tailβ€˜π·))
3533, 34sylan 581 . . . . . . . . . . . 12 ((𝐷 ∈ DirRel ∧ 𝑀 ∈ 𝑋) β†’ ((tailβ€˜π·)β€˜π‘€) ∈ ran (tailβ€˜π·))
3635ad2ant2r 746 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ ((tailβ€˜π·)β€˜π‘€) ∈ ran (tailβ€˜π·))
37 dirtr 18555 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ π‘₯ ∈ V) ∧ (𝑒𝐷𝑀 ∧ 𝑀𝐷π‘₯)) β†’ 𝑒𝐷π‘₯)
3837exp32 422 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ π‘₯ ∈ V) β†’ (𝑒𝐷𝑀 β†’ (𝑀𝐷π‘₯ β†’ 𝑒𝐷π‘₯)))
3938elvd 3482 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel β†’ (𝑒𝐷𝑀 β†’ (𝑀𝐷π‘₯ β†’ 𝑒𝐷π‘₯)))
4039com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel β†’ (𝑀𝐷π‘₯ β†’ (𝑒𝐷𝑀 β†’ 𝑒𝐷π‘₯)))
4140imp 408 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑀𝐷π‘₯) β†’ (𝑒𝐷𝑀 β†’ 𝑒𝐷π‘₯))
4241ad2ant2rl 748 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ 𝑀𝐷π‘₯)) β†’ (𝑒𝐷𝑀 β†’ 𝑒𝐷π‘₯))
43 dirtr 18555 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐷 ∈ DirRel ∧ π‘₯ ∈ V) ∧ (𝑣𝐷𝑀 ∧ 𝑀𝐷π‘₯)) β†’ 𝑣𝐷π‘₯)
4443exp32 422 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ DirRel ∧ π‘₯ ∈ V) β†’ (𝑣𝐷𝑀 β†’ (𝑀𝐷π‘₯ β†’ 𝑣𝐷π‘₯)))
4544elvd 3482 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ DirRel β†’ (𝑣𝐷𝑀 β†’ (𝑀𝐷π‘₯ β†’ 𝑣𝐷π‘₯)))
4645com23 86 . . . . . . . . . . . . . . . . . . . 20 (𝐷 ∈ DirRel β†’ (𝑀𝐷π‘₯ β†’ (𝑣𝐷𝑀 β†’ 𝑣𝐷π‘₯)))
4746imp 408 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ DirRel ∧ 𝑀𝐷π‘₯) β†’ (𝑣𝐷𝑀 β†’ 𝑣𝐷π‘₯))
4847ad2ant2rl 748 . . . . . . . . . . . . . . . . . 18 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ 𝑀𝐷π‘₯)) β†’ (𝑣𝐷𝑀 β†’ 𝑣𝐷π‘₯))
4942, 48anim12d 610 . . . . . . . . . . . . . . . . 17 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ 𝑀𝐷π‘₯)) β†’ ((𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀) β†’ (𝑒𝐷π‘₯ ∧ 𝑣𝐷π‘₯)))
5049expr 458 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ 𝑀 ∈ 𝑋) β†’ (𝑀𝐷π‘₯ β†’ ((𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀) β†’ (𝑒𝐷π‘₯ ∧ 𝑣𝐷π‘₯))))
5150com23 86 . . . . . . . . . . . . . . 15 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ 𝑀 ∈ 𝑋) β†’ ((𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀) β†’ (𝑀𝐷π‘₯ β†’ (𝑒𝐷π‘₯ ∧ 𝑣𝐷π‘₯))))
5251impr 456 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ (𝑀𝐷π‘₯ β†’ (𝑒𝐷π‘₯ ∧ 𝑣𝐷π‘₯)))
53 vex 3479 . . . . . . . . . . . . . . . 16 π‘₯ ∈ V
541eltail 35307 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ 𝑀 ∈ 𝑋 ∧ π‘₯ ∈ V) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘€) ↔ 𝑀𝐷π‘₯))
5553, 54mp3an3 1451 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ 𝑀 ∈ 𝑋) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘€) ↔ 𝑀𝐷π‘₯))
5655ad2ant2r 746 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘€) ↔ 𝑀𝐷π‘₯))
571eltail 35307 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑒 ∈ 𝑋 ∧ π‘₯ ∈ V) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ↔ 𝑒𝐷π‘₯))
5853, 57mp3an3 1451 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑒 ∈ 𝑋) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ↔ 𝑒𝐷π‘₯))
5958adantrr 716 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ↔ 𝑒𝐷π‘₯))
601eltail 35307 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ DirRel ∧ 𝑣 ∈ 𝑋 ∧ π‘₯ ∈ V) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£) ↔ 𝑣𝐷π‘₯))
6153, 60mp3an3 1451 . . . . . . . . . . . . . . . . 17 ((𝐷 ∈ DirRel ∧ 𝑣 ∈ 𝑋) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£) ↔ 𝑣𝐷π‘₯))
6261adantrl 715 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£) ↔ 𝑣𝐷π‘₯))
6359, 62anbi12d 632 . . . . . . . . . . . . . . 15 ((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ ((π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ∧ π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£)) ↔ (𝑒𝐷π‘₯ ∧ 𝑣𝐷π‘₯)))
6463adantr 482 . . . . . . . . . . . . . 14 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ ((π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ∧ π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£)) ↔ (𝑒𝐷π‘₯ ∧ 𝑣𝐷π‘₯)))
6552, 56, 643imtr4d 294 . . . . . . . . . . . . 13 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘€) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ∧ π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£))))
66 elin 3965 . . . . . . . . . . . . 13 (π‘₯ ∈ (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘’) ∧ π‘₯ ∈ ((tailβ€˜π·)β€˜π‘£)))
6765, 66syl6ibr 252 . . . . . . . . . . . 12 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ (π‘₯ ∈ ((tailβ€˜π·)β€˜π‘€) β†’ π‘₯ ∈ (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£))))
6867ssrdv 3989 . . . . . . . . . . 11 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ ((tailβ€˜π·)β€˜π‘€) βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)))
69 sseq1 4008 . . . . . . . . . . . 12 (𝑧 = ((tailβ€˜π·)β€˜π‘€) β†’ (𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ ((tailβ€˜π·)β€˜π‘€) βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£))))
7069rspcev 3613 . . . . . . . . . . 11 ((((tailβ€˜π·)β€˜π‘€) ∈ ran (tailβ€˜π·) ∧ ((tailβ€˜π·)β€˜π‘€) βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£))) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)))
7136, 68, 70syl2anc 585 . . . . . . . . . 10 (((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) ∧ (𝑀 ∈ 𝑋 ∧ (𝑒𝐷𝑀 ∧ 𝑣𝐷𝑀))) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)))
7232, 71rexlimddv 3162 . . . . . . . . 9 ((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)))
73 ineq1 4206 . . . . . . . . . . . 12 (((tailβ€˜π·)β€˜π‘’) = π‘₯ β†’ (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)) = (π‘₯ ∩ ((tailβ€˜π·)β€˜π‘£)))
7473sseq2d 4015 . . . . . . . . . . 11 (((tailβ€˜π·)β€˜π‘’) = π‘₯ β†’ (𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ 𝑧 βŠ† (π‘₯ ∩ ((tailβ€˜π·)β€˜π‘£))))
7574rexbidv 3179 . . . . . . . . . 10 (((tailβ€˜π·)β€˜π‘’) = π‘₯ β†’ (βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ ((tailβ€˜π·)β€˜π‘£))))
76 ineq2 4207 . . . . . . . . . . . 12 (((tailβ€˜π·)β€˜π‘£) = 𝑦 β†’ (π‘₯ ∩ ((tailβ€˜π·)β€˜π‘£)) = (π‘₯ ∩ 𝑦))
7776sseq2d 4015 . . . . . . . . . . 11 (((tailβ€˜π·)β€˜π‘£) = 𝑦 β†’ (𝑧 βŠ† (π‘₯ ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ 𝑧 βŠ† (π‘₯ ∩ 𝑦)))
7877rexbidv 3179 . . . . . . . . . 10 (((tailβ€˜π·)β€˜π‘£) = 𝑦 β†’ (βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
7975, 78sylan9bb 511 . . . . . . . . 9 ((((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ ((tailβ€˜π·)β€˜π‘£) = 𝑦) β†’ (βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (((tailβ€˜π·)β€˜π‘’) ∩ ((tailβ€˜π·)β€˜π‘£)) ↔ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
8072, 79syl5ibcom 244 . . . . . . . 8 ((𝐷 ∈ DirRel ∧ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) β†’ ((((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ ((tailβ€˜π·)β€˜π‘£) = 𝑦) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
8180rexlimdvva 3212 . . . . . . 7 (𝐷 ∈ DirRel β†’ (βˆƒπ‘’ ∈ 𝑋 βˆƒπ‘£ ∈ 𝑋 (((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ ((tailβ€˜π·)β€˜π‘£) = 𝑦) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
8230, 81biimtrrid 242 . . . . . 6 (𝐷 ∈ DirRel β†’ ((βˆƒπ‘’ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘’) = π‘₯ ∧ βˆƒπ‘£ ∈ 𝑋 ((tailβ€˜π·)β€˜π‘£) = 𝑦) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
8329, 82sylbid 239 . . . . 5 (𝐷 ∈ DirRel β†’ ((π‘₯ ∈ ran (tailβ€˜π·) ∧ 𝑦 ∈ ran (tailβ€˜π·)) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
8483adantr 482 . . . 4 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ((π‘₯ ∈ ran (tailβ€˜π·) ∧ 𝑦 ∈ ran (tailβ€˜π·)) β†’ βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
8584ralrimivv 3199 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ βˆ€π‘₯ ∈ ran (tailβ€˜π·)βˆ€π‘¦ ∈ ran (tailβ€˜π·)βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦))
8614, 25, 853jca 1129 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ (ran (tailβ€˜π·) β‰  βˆ… ∧ βˆ… βˆ‰ ran (tailβ€˜π·) ∧ βˆ€π‘₯ ∈ ran (tailβ€˜π·)βˆ€π‘¦ ∈ ran (tailβ€˜π·)βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))
87 dmexg 7894 . . . . 5 (𝐷 ∈ DirRel β†’ dom 𝐷 ∈ V)
881, 87eqeltrid 2838 . . . 4 (𝐷 ∈ DirRel β†’ 𝑋 ∈ V)
8988adantr 482 . . 3 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ 𝑋 ∈ V)
90 isfbas2 23339 . . 3 (𝑋 ∈ V β†’ (ran (tailβ€˜π·) ∈ (fBasβ€˜π‘‹) ↔ (ran (tailβ€˜π·) βŠ† 𝒫 𝑋 ∧ (ran (tailβ€˜π·) β‰  βˆ… ∧ βˆ… βˆ‰ ran (tailβ€˜π·) ∧ βˆ€π‘₯ ∈ ran (tailβ€˜π·)βˆ€π‘¦ ∈ ran (tailβ€˜π·)βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))))
9189, 90syl 17 . 2 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ (ran (tailβ€˜π·) ∈ (fBasβ€˜π‘‹) ↔ (ran (tailβ€˜π·) βŠ† 𝒫 𝑋 ∧ (ran (tailβ€˜π·) β‰  βˆ… ∧ βˆ… βˆ‰ ran (tailβ€˜π·) ∧ βˆ€π‘₯ ∈ ran (tailβ€˜π·)βˆ€π‘¦ ∈ ran (tailβ€˜π·)βˆƒπ‘§ ∈ ran (tailβ€˜π·)𝑧 βŠ† (π‘₯ ∩ 𝑦)))))
924, 86, 91mpbir2and 712 1 ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ran (tailβ€˜π·) ∈ (fBasβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941   βˆ‰ wnel 3047  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603   class class class wbr 5149  dom cdm 5677  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  DirRelcdir 18547  tailctail 18548  fBascfbas 20932
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-dir 18549  df-tail 18550  df-fbas 20941
This theorem is referenced by:  filnetlem4  35314
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