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Theorem tsmsfbas 23495
Description: The collection of all sets of the form 𝐹(𝑧) = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}, which can be read as the set of all finite subsets of 𝐴 which contain 𝑧 as a subset, for each finite subset 𝑧 of 𝐴, form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tsmsfbas.s 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsfbas.f 𝐹 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
tsmsfbas.l 𝐿 = ran 𝐹
tsmsfbas.a (πœ‘ β†’ 𝐴 ∈ π‘Š)
Assertion
Ref Expression
tsmsfbas (πœ‘ β†’ 𝐿 ∈ (fBasβ€˜π‘†))
Distinct variable groups:   𝑧,𝐴   𝑦,𝑧,𝑆
Allowed substitution hints:   πœ‘(𝑦,𝑧)   𝐴(𝑦)   𝐹(𝑦,𝑧)   𝐿(𝑦,𝑧)   π‘Š(𝑦,𝑧)

Proof of Theorem tsmsfbas
Dummy variables 𝑒 π‘Ž 𝑣 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmsfbas.a . 2 (πœ‘ β†’ 𝐴 ∈ π‘Š)
2 elex 3466 . 2 (𝐴 ∈ π‘Š β†’ 𝐴 ∈ V)
3 tsmsfbas.l . . 3 𝐿 = ran 𝐹
4 ssrab2 4042 . . . . . . 7 {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} βŠ† 𝑆
5 tsmsfbas.s . . . . . . . . . 10 𝑆 = (𝒫 𝐴 ∩ Fin)
6 pwexg 5338 . . . . . . . . . . 11 (𝐴 ∈ V β†’ 𝒫 𝐴 ∈ V)
7 inex1g 5281 . . . . . . . . . . 11 (𝒫 𝐴 ∈ V β†’ (𝒫 𝐴 ∩ Fin) ∈ V)
86, 7syl 17 . . . . . . . . . 10 (𝐴 ∈ V β†’ (𝒫 𝐴 ∩ Fin) ∈ V)
95, 8eqeltrid 2842 . . . . . . . . 9 (𝐴 ∈ V β†’ 𝑆 ∈ V)
109adantr 482 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ 𝑆 ∈ V)
11 elpw2g 5306 . . . . . . . 8 (𝑆 ∈ V β†’ ({𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} βŠ† 𝑆))
1210, 11syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ ({𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} βŠ† 𝑆))
134, 12mpbiri 258 . . . . . 6 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ∈ 𝒫 𝑆)
14 tsmsfbas.f . . . . . 6 𝐹 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
1513, 14fmptd 7067 . . . . 5 (𝐴 ∈ V β†’ 𝐹:π‘†βŸΆπ’« 𝑆)
1615frnd 6681 . . . 4 (𝐴 ∈ V β†’ ran 𝐹 βŠ† 𝒫 𝑆)
17 0ss 4361 . . . . . . . . . 10 βˆ… βŠ† 𝐴
18 0fin 9122 . . . . . . . . . 10 βˆ… ∈ Fin
19 elfpw 9305 . . . . . . . . . 10 (βˆ… ∈ (𝒫 𝐴 ∩ Fin) ↔ (βˆ… βŠ† 𝐴 ∧ βˆ… ∈ Fin))
2017, 18, 19mpbir2an 710 . . . . . . . . 9 βˆ… ∈ (𝒫 𝐴 ∩ Fin)
2120, 5eleqtrri 2837 . . . . . . . 8 βˆ… ∈ 𝑆
22 0ss 4361 . . . . . . . . 9 βˆ… βŠ† 𝑦
2322rgenw 3069 . . . . . . . 8 βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦
24 rabid2 3439 . . . . . . . . . 10 (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
25 sseq1 3974 . . . . . . . . . . 11 (𝑧 = βˆ… β†’ (𝑧 βŠ† 𝑦 ↔ βˆ… βŠ† 𝑦))
2625ralbidv 3175 . . . . . . . . . 10 (𝑧 = βˆ… β†’ (βˆ€π‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦 ↔ βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦))
2724, 26bitrid 283 . . . . . . . . 9 (𝑧 = βˆ… β†’ (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦))
2827rspcev 3584 . . . . . . . 8 ((βˆ… ∈ 𝑆 ∧ βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
2921, 23, 28mp2an 691 . . . . . . 7 βˆƒπ‘§ ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}
3014elrnmpt 5916 . . . . . . . 8 (𝑆 ∈ V β†’ (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}))
319, 30syl 17 . . . . . . 7 (𝐴 ∈ V β†’ (𝑆 ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}))
3229, 31mpbiri 258 . . . . . 6 (𝐴 ∈ V β†’ 𝑆 ∈ ran 𝐹)
3332ne0d 4300 . . . . 5 (𝐴 ∈ V β†’ ran 𝐹 β‰  βˆ…)
34 simpr 486 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 ∈ 𝑆)
35 ssid 3971 . . . . . . . . . . . 12 𝑧 βŠ† 𝑧
36 sseq2 3975 . . . . . . . . . . . . 13 (𝑦 = 𝑧 β†’ (𝑧 βŠ† 𝑦 ↔ 𝑧 βŠ† 𝑧))
3736rspcev 3584 . . . . . . . . . . . 12 ((𝑧 ∈ 𝑆 ∧ 𝑧 βŠ† 𝑧) β†’ βˆƒπ‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
3834, 35, 37sylancl 587 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ βˆƒπ‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
39 rabn0 4350 . . . . . . . . . . 11 ({𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} β‰  βˆ… ↔ βˆƒπ‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
4038, 39sylibr 233 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} β‰  βˆ…)
4140necomd 3000 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ βˆ… β‰  {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
4241neneqd 2949 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ Β¬ βˆ… = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
4342nrexdv 3147 . . . . . . 7 (𝐴 ∈ V β†’ Β¬ βˆƒπ‘§ ∈ 𝑆 βˆ… = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
44 0ex 5269 . . . . . . . 8 βˆ… ∈ V
4514elrnmpt 5916 . . . . . . . 8 (βˆ… ∈ V β†’ (βˆ… ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ 𝑆 βˆ… = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}))
4644, 45ax-mp 5 . . . . . . 7 (βˆ… ∈ ran 𝐹 ↔ βˆƒπ‘§ ∈ 𝑆 βˆ… = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
4743, 46sylnibr 329 . . . . . 6 (𝐴 ∈ V β†’ Β¬ βˆ… ∈ ran 𝐹)
48 df-nel 3051 . . . . . 6 (βˆ… βˆ‰ ran 𝐹 ↔ Β¬ βˆ… ∈ ran 𝐹)
4947, 48sylibr 233 . . . . 5 (𝐴 ∈ V β†’ βˆ… βˆ‰ ran 𝐹)
50 elfpw 9305 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑒 βŠ† 𝐴 ∧ 𝑒 ∈ Fin))
5150simplbi 499 . . . . . . . . . . . . . . . . 17 (𝑒 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑒 βŠ† 𝐴)
5251, 5eleq2s 2856 . . . . . . . . . . . . . . . 16 (𝑒 ∈ 𝑆 β†’ 𝑒 βŠ† 𝐴)
53 elfpw 9305 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣 βŠ† 𝐴 ∧ 𝑣 ∈ Fin))
5453simplbi 499 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑣 βŠ† 𝐴)
5554, 5eleq2s 2856 . . . . . . . . . . . . . . . 16 (𝑣 ∈ 𝑆 β†’ 𝑣 βŠ† 𝐴)
5652, 55anim12i 614 . . . . . . . . . . . . . . 15 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βŠ† 𝐴 ∧ 𝑣 βŠ† 𝐴))
57 unss 4149 . . . . . . . . . . . . . . 15 ((𝑒 βŠ† 𝐴 ∧ 𝑣 βŠ† 𝐴) ↔ (𝑒 βˆͺ 𝑣) βŠ† 𝐴)
5856, 57sylib 217 . . . . . . . . . . . . . 14 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βˆͺ 𝑣) βŠ† 𝐴)
59 elinel2 4161 . . . . . . . . . . . . . . . 16 (𝑒 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑒 ∈ Fin)
6059, 5eleq2s 2856 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝑆 β†’ 𝑒 ∈ Fin)
61 elinel2 4161 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑣 ∈ Fin)
6261, 5eleq2s 2856 . . . . . . . . . . . . . . 15 (𝑣 ∈ 𝑆 β†’ 𝑣 ∈ Fin)
63 unfi 9123 . . . . . . . . . . . . . . 15 ((𝑒 ∈ Fin ∧ 𝑣 ∈ Fin) β†’ (𝑒 βˆͺ 𝑣) ∈ Fin)
6460, 62, 63syl2an 597 . . . . . . . . . . . . . 14 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βˆͺ 𝑣) ∈ Fin)
65 elfpw 9305 . . . . . . . . . . . . . 14 ((𝑒 βˆͺ 𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑒 βˆͺ 𝑣) βŠ† 𝐴 ∧ (𝑒 βˆͺ 𝑣) ∈ Fin))
6658, 64, 65sylanbrc 584 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βˆͺ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
6766adantl 483 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ (𝑒 βˆͺ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
6867, 5eleqtrrdi 2849 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ (𝑒 βˆͺ 𝑣) ∈ 𝑆)
69 eqidd 2738 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
70 sseq1 3974 . . . . . . . . . . . . 13 (π‘Ž = (𝑒 βˆͺ 𝑣) β†’ (π‘Ž βŠ† 𝑦 ↔ (𝑒 βˆͺ 𝑣) βŠ† 𝑦))
7170rabbidv 3418 . . . . . . . . . . . 12 (π‘Ž = (𝑒 βˆͺ 𝑣) β†’ {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
7271rspceeqv 3600 . . . . . . . . . . 11 (((𝑒 βˆͺ 𝑣) ∈ 𝑆 ∧ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β†’ βˆƒπ‘Ž ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
7368, 69, 72syl2anc 585 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ βˆƒπ‘Ž ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
749adantr 482 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ 𝑆 ∈ V)
75 rabexg 5293 . . . . . . . . . . . 12 (𝑆 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ V)
7674, 75syl 17 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ V)
77 sseq1 3974 . . . . . . . . . . . . . . 15 (𝑧 = π‘Ž β†’ (𝑧 βŠ† 𝑦 ↔ π‘Ž βŠ† 𝑦))
7877rabbidv 3418 . . . . . . . . . . . . . 14 (𝑧 = π‘Ž β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
7978cbvmptv 5223 . . . . . . . . . . . . 13 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}) = (π‘Ž ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
8014, 79eqtri 2765 . . . . . . . . . . . 12 𝐹 = (π‘Ž ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
8180elrnmpt 5916 . . . . . . . . . . 11 ({𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ V β†’ ({𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ ran 𝐹 ↔ βˆƒπ‘Ž ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦}))
8276, 81syl 17 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ ({𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ ran 𝐹 ↔ βˆƒπ‘Ž ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦}))
8373, 82mpbird 257 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ ran 𝐹)
84 pwidg 4585 . . . . . . . . . 10 ({𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ V β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
8576, 84syl 17 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
86 inelcm 4429 . . . . . . . . 9 (({𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ ran 𝐹 ∧ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β†’ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…)
8783, 85, 86syl2anc 585 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…)
8887ralrimivva 3198 . . . . . . 7 (𝐴 ∈ V β†’ βˆ€π‘’ ∈ 𝑆 βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…)
89 rabexg 5293 . . . . . . . . . 10 (𝑆 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V)
909, 89syl 17 . . . . . . . . 9 (𝐴 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V)
9190ralrimivw 3148 . . . . . . . 8 (𝐴 ∈ V β†’ βˆ€π‘’ ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V)
92 sseq1 3974 . . . . . . . . . . . 12 (𝑧 = 𝑒 β†’ (𝑧 βŠ† 𝑦 ↔ 𝑒 βŠ† 𝑦))
9392rabbidv 3418 . . . . . . . . . . 11 (𝑧 = 𝑒 β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦})
9493cbvmptv 5223 . . . . . . . . . 10 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}) = (𝑒 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦})
9514, 94eqtri 2765 . . . . . . . . 9 𝐹 = (𝑒 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦})
96 ineq1 4170 . . . . . . . . . . . . . 14 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = ({𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}))
97 inrab 4271 . . . . . . . . . . . . . . 15 ({𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑒 βŠ† 𝑦 ∧ 𝑣 βŠ† 𝑦)}
98 unss 4149 . . . . . . . . . . . . . . . 16 ((𝑒 βŠ† 𝑦 ∧ 𝑣 βŠ† 𝑦) ↔ (𝑒 βˆͺ 𝑣) βŠ† 𝑦)
9998rabbii 3416 . . . . . . . . . . . . . . 15 {𝑦 ∈ 𝑆 ∣ (𝑒 βŠ† 𝑦 ∧ 𝑣 βŠ† 𝑦)} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}
10097, 99eqtri 2765 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}
10196, 100eqtrdi 2793 . . . . . . . . . . . . 13 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
102101pweqd 4582 . . . . . . . . . . . 12 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
103102ineq2d 4177 . . . . . . . . . . 11 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) = (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}))
104103neeq1d 3004 . . . . . . . . . 10 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ ((ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ… ↔ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…))
105104ralbidv 3175 . . . . . . . . 9 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ (βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ… ↔ βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…))
10695, 105ralrnmptw 7049 . . . . . . . 8 (βˆ€π‘’ ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V β†’ (βˆ€π‘Ž ∈ ran πΉβˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ… ↔ βˆ€π‘’ ∈ 𝑆 βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…))
10791, 106syl 17 . . . . . . 7 (𝐴 ∈ V β†’ (βˆ€π‘Ž ∈ ran πΉβˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ… ↔ βˆ€π‘’ ∈ 𝑆 βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…))
10888, 107mpbird 257 . . . . . 6 (𝐴 ∈ V β†’ βˆ€π‘Ž ∈ ran πΉβˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ…)
109 rabexg 5293 . . . . . . . . . 10 (𝑆 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V)
1109, 109syl 17 . . . . . . . . 9 (𝐴 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V)
111110ralrimivw 3148 . . . . . . . 8 (𝐴 ∈ V β†’ βˆ€π‘£ ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V)
112 sseq1 3974 . . . . . . . . . . . 12 (𝑧 = 𝑣 β†’ (𝑧 βŠ† 𝑦 ↔ 𝑣 βŠ† 𝑦))
113112rabbidv 3418 . . . . . . . . . . 11 (𝑧 = 𝑣 β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})
114113cbvmptv 5223 . . . . . . . . . 10 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}) = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})
11514, 114eqtri 2765 . . . . . . . . 9 𝐹 = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})
116 ineq2 4171 . . . . . . . . . . . 12 (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} β†’ (π‘Ž ∩ 𝑏) = (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}))
117116pweqd 4582 . . . . . . . . . . 11 (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} β†’ 𝒫 (π‘Ž ∩ 𝑏) = 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}))
118117ineq2d 4177 . . . . . . . . . 10 (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} β†’ (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) = (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})))
119118neeq1d 3004 . . . . . . . . 9 (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} β†’ ((ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ… ↔ (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ…))
120115, 119ralrnmptw 7049 . . . . . . . 8 (βˆ€π‘£ ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V β†’ (βˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ… ↔ βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ…))
121111, 120syl 17 . . . . . . 7 (𝐴 ∈ V β†’ (βˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ… ↔ βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ…))
122121ralbidv 3175 . . . . . 6 (𝐴 ∈ V β†’ (βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ… ↔ βˆ€π‘Ž ∈ ran πΉβˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})) β‰  βˆ…))
123108, 122mpbird 257 . . . . 5 (𝐴 ∈ V β†’ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ…)
12433, 49, 1233jca 1129 . . . 4 (𝐴 ∈ V β†’ (ran 𝐹 β‰  βˆ… ∧ βˆ… βˆ‰ ran 𝐹 ∧ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ…))
125 isfbas 23196 . . . . 5 (𝑆 ∈ V β†’ (ran 𝐹 ∈ (fBasβ€˜π‘†) ↔ (ran 𝐹 βŠ† 𝒫 𝑆 ∧ (ran 𝐹 β‰  βˆ… ∧ βˆ… βˆ‰ ran 𝐹 ∧ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ…))))
1269, 125syl 17 . . . 4 (𝐴 ∈ V β†’ (ran 𝐹 ∈ (fBasβ€˜π‘†) ↔ (ran 𝐹 βŠ† 𝒫 𝑆 ∧ (ran 𝐹 β‰  βˆ… ∧ βˆ… βˆ‰ ran 𝐹 ∧ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ…))))
12716, 124, 126mpbir2and 712 . . 3 (𝐴 ∈ V β†’ ran 𝐹 ∈ (fBasβ€˜π‘†))
1283, 127eqeltrid 2842 . 2 (𝐴 ∈ V β†’ 𝐿 ∈ (fBasβ€˜π‘†))
1291, 2, 1283syl 18 1 (πœ‘ β†’ 𝐿 ∈ (fBasβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ‰ wnel 3050  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆͺ cun 3913   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565   ↦ cmpt 5193  ran crn 5639  β€˜cfv 6501  Fincfn 8890  fBascfbas 20800
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-en 8891  df-fin 8894  df-fbas 20809
This theorem is referenced by:  eltsms  23500  haustsms  23503  tsmscls  23505  tsmsmhm  23513  tsmsadd  23514
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