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Theorem tsmsfbas 23639
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 3492 . 2 (𝐴 ∈ π‘Š β†’ 𝐴 ∈ V)
3 tsmsfbas.l . . 3 𝐿 = ran 𝐹
4 ssrab2 4077 . . . . . . 7 {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} βŠ† 𝑆
5 tsmsfbas.s . . . . . . . . . 10 𝑆 = (𝒫 𝐴 ∩ Fin)
6 pwexg 5376 . . . . . . . . . . 11 (𝐴 ∈ V β†’ 𝒫 𝐴 ∈ V)
7 inex1g 5319 . . . . . . . . . . 11 (𝒫 𝐴 ∈ V β†’ (𝒫 𝐴 ∩ Fin) ∈ V)
86, 7syl 17 . . . . . . . . . 10 (𝐴 ∈ V β†’ (𝒫 𝐴 ∩ Fin) ∈ V)
95, 8eqeltrid 2837 . . . . . . . . 9 (𝐴 ∈ V β†’ 𝑆 ∈ V)
109adantr 481 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ 𝑆 ∈ V)
11 elpw2g 5344 . . . . . . . 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 257 . . . . . 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 7115 . . . . 5 (𝐴 ∈ V β†’ 𝐹:π‘†βŸΆπ’« 𝑆)
1615frnd 6725 . . . 4 (𝐴 ∈ V β†’ ran 𝐹 βŠ† 𝒫 𝑆)
17 0ss 4396 . . . . . . . . . 10 βˆ… βŠ† 𝐴
18 0fin 9173 . . . . . . . . . 10 βˆ… ∈ Fin
19 elfpw 9356 . . . . . . . . . 10 (βˆ… ∈ (𝒫 𝐴 ∩ Fin) ↔ (βˆ… βŠ† 𝐴 ∧ βˆ… ∈ Fin))
2017, 18, 19mpbir2an 709 . . . . . . . . 9 βˆ… ∈ (𝒫 𝐴 ∩ Fin)
2120, 5eleqtrri 2832 . . . . . . . 8 βˆ… ∈ 𝑆
22 0ss 4396 . . . . . . . . 9 βˆ… βŠ† 𝑦
2322rgenw 3065 . . . . . . . 8 βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦
24 rabid2 3464 . . . . . . . . . 10 (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
25 sseq1 4007 . . . . . . . . . . 11 (𝑧 = βˆ… β†’ (𝑧 βŠ† 𝑦 ↔ βˆ… βŠ† 𝑦))
2625ralbidv 3177 . . . . . . . . . 10 (𝑧 = βˆ… β†’ (βˆ€π‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦 ↔ βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦))
2724, 26bitrid 282 . . . . . . . . 9 (𝑧 = βˆ… β†’ (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} ↔ βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦))
2827rspcev 3612 . . . . . . . 8 ((βˆ… ∈ 𝑆 ∧ βˆ€π‘¦ ∈ 𝑆 βˆ… βŠ† 𝑦) β†’ βˆƒπ‘§ ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
2921, 23, 28mp2an 690 . . . . . . 7 βˆƒπ‘§ ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}
3014elrnmpt 5955 . . . . . . . 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 257 . . . . . 6 (𝐴 ∈ V β†’ 𝑆 ∈ ran 𝐹)
3332ne0d 4335 . . . . 5 (𝐴 ∈ V β†’ ran 𝐹 β‰  βˆ…)
34 simpr 485 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ 𝑧 ∈ 𝑆)
35 ssid 4004 . . . . . . . . . . . 12 𝑧 βŠ† 𝑧
36 sseq2 4008 . . . . . . . . . . . . 13 (𝑦 = 𝑧 β†’ (𝑧 βŠ† 𝑦 ↔ 𝑧 βŠ† 𝑧))
3736rspcev 3612 . . . . . . . . . . . 12 ((𝑧 ∈ 𝑆 ∧ 𝑧 βŠ† 𝑧) β†’ βˆƒπ‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
3834, 35, 37sylancl 586 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ βˆƒπ‘¦ ∈ 𝑆 𝑧 βŠ† 𝑦)
39 rabn0 4385 . . . . . . . . . . 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 2996 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ βˆ… β‰  {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
4241neneqd 2945 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) β†’ Β¬ βˆ… = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
4342nrexdv 3149 . . . . . . 7 (𝐴 ∈ V β†’ Β¬ βˆƒπ‘§ ∈ 𝑆 βˆ… = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
44 0ex 5307 . . . . . . . 8 βˆ… ∈ V
4514elrnmpt 5955 . . . . . . . 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 328 . . . . . 6 (𝐴 ∈ V β†’ Β¬ βˆ… ∈ ran 𝐹)
48 df-nel 3047 . . . . . 6 (βˆ… βˆ‰ ran 𝐹 ↔ Β¬ βˆ… ∈ ran 𝐹)
4947, 48sylibr 233 . . . . 5 (𝐴 ∈ V β†’ βˆ… βˆ‰ ran 𝐹)
50 elfpw 9356 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑒 βŠ† 𝐴 ∧ 𝑒 ∈ Fin))
5150simplbi 498 . . . . . . . . . . . . . . . . 17 (𝑒 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑒 βŠ† 𝐴)
5251, 5eleq2s 2851 . . . . . . . . . . . . . . . 16 (𝑒 ∈ 𝑆 β†’ 𝑒 βŠ† 𝐴)
53 elfpw 9356 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣 βŠ† 𝐴 ∧ 𝑣 ∈ Fin))
5453simplbi 498 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑣 βŠ† 𝐴)
5554, 5eleq2s 2851 . . . . . . . . . . . . . . . 16 (𝑣 ∈ 𝑆 β†’ 𝑣 βŠ† 𝐴)
5652, 55anim12i 613 . . . . . . . . . . . . . . 15 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βŠ† 𝐴 ∧ 𝑣 βŠ† 𝐴))
57 unss 4184 . . . . . . . . . . . . . . 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 4196 . . . . . . . . . . . . . . . 16 (𝑒 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑒 ∈ Fin)
6059, 5eleq2s 2851 . . . . . . . . . . . . . . 15 (𝑒 ∈ 𝑆 β†’ 𝑒 ∈ Fin)
61 elinel2 4196 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑣 ∈ Fin)
6261, 5eleq2s 2851 . . . . . . . . . . . . . . 15 (𝑣 ∈ 𝑆 β†’ 𝑣 ∈ Fin)
63 unfi 9174 . . . . . . . . . . . . . . 15 ((𝑒 ∈ Fin ∧ 𝑣 ∈ Fin) β†’ (𝑒 βˆͺ 𝑣) ∈ Fin)
6460, 62, 63syl2an 596 . . . . . . . . . . . . . 14 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βˆͺ 𝑣) ∈ Fin)
65 elfpw 9356 . . . . . . . . . . . . . 14 ((𝑒 βˆͺ 𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑒 βˆͺ 𝑣) βŠ† 𝐴 ∧ (𝑒 βˆͺ 𝑣) ∈ Fin))
6658, 64, 65sylanbrc 583 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) β†’ (𝑒 βˆͺ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
6766adantl 482 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ (𝑒 βˆͺ 𝑣) ∈ (𝒫 𝐴 ∩ Fin))
6867, 5eleqtrrdi 2844 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ (𝑒 βˆͺ 𝑣) ∈ 𝑆)
69 eqidd 2733 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
70 sseq1 4007 . . . . . . . . . . . . 13 (π‘Ž = (𝑒 βˆͺ 𝑣) β†’ (π‘Ž βŠ† 𝑦 ↔ (𝑒 βˆͺ 𝑣) βŠ† 𝑦))
7170rabbidv 3440 . . . . . . . . . . . 12 (π‘Ž = (𝑒 βˆͺ 𝑣) β†’ {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
7271rspceeqv 3633 . . . . . . . . . . 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 584 . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ 𝑆 ∈ V)
75 rabexg 5331 . . . . . . . . . . . 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 4007 . . . . . . . . . . . . . . 15 (𝑧 = π‘Ž β†’ (𝑧 βŠ† 𝑦 ↔ π‘Ž βŠ† 𝑦))
7877rabbidv 3440 . . . . . . . . . . . . . 14 (𝑧 = π‘Ž β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
7978cbvmptv 5261 . . . . . . . . . . . . 13 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}) = (π‘Ž ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
8014, 79eqtri 2760 . . . . . . . . . . . 12 𝐹 = (π‘Ž ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ π‘Ž βŠ† 𝑦})
8180elrnmpt 5955 . . . . . . . . . . 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 256 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦} ∈ ran 𝐹)
84 pwidg 4622 . . . . . . . . . 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 4464 . . . . . . . . 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 584 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑒 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) β†’ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…)
8887ralrimivva 3200 . . . . . . 7 (𝐴 ∈ V β†’ βˆ€π‘’ ∈ 𝑆 βˆ€π‘£ ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}) β‰  βˆ…)
89 rabexg 5331 . . . . . . . . . 10 (𝑆 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V)
909, 89syl 17 . . . . . . . . 9 (𝐴 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V)
9190ralrimivw 3150 . . . . . . . 8 (𝐴 ∈ V β†’ βˆ€π‘’ ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∈ V)
92 sseq1 4007 . . . . . . . . . . . 12 (𝑧 = 𝑒 β†’ (𝑧 βŠ† 𝑦 ↔ 𝑒 βŠ† 𝑦))
9392rabbidv 3440 . . . . . . . . . . 11 (𝑧 = 𝑒 β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦})
9493cbvmptv 5261 . . . . . . . . . 10 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}) = (𝑒 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦})
9514, 94eqtri 2760 . . . . . . . . 9 𝐹 = (𝑒 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦})
96 ineq1 4205 . . . . . . . . . . . . . 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 4306 . . . . . . . . . . . . . . 15 ({𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑒 βŠ† 𝑦 ∧ 𝑣 βŠ† 𝑦)}
98 unss 4184 . . . . . . . . . . . . . . . 16 ((𝑒 βŠ† 𝑦 ∧ 𝑣 βŠ† 𝑦) ↔ (𝑒 βˆͺ 𝑣) βŠ† 𝑦)
9998rabbii 3438 . . . . . . . . . . . . . . 15 {𝑦 ∈ 𝑆 ∣ (𝑒 βŠ† 𝑦 ∧ 𝑣 βŠ† 𝑦)} = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}
10097, 99eqtri 2760 . . . . . . . . . . . . . 14 ({𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦}
10196, 100eqtrdi 2788 . . . . . . . . . . . . 13 (π‘Ž = {𝑦 ∈ 𝑆 ∣ 𝑒 βŠ† 𝑦} β†’ (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑒 βˆͺ 𝑣) βŠ† 𝑦})
102101pweqd 4619 . . . . . . . . . . . 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 4212 . . . . . . . . . . 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 3000 . . . . . . . . . 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 3177 . . . . . . . . 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 7095 . . . . . . . 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 256 . . . . . 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 5331 . . . . . . . . . 10 (𝑆 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V)
1109, 109syl 17 . . . . . . . . 9 (𝐴 ∈ V β†’ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V)
111110ralrimivw 3150 . . . . . . . 8 (𝐴 ∈ V β†’ βˆ€π‘£ ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} ∈ V)
112 sseq1 4007 . . . . . . . . . . . 12 (𝑧 = 𝑣 β†’ (𝑧 βŠ† 𝑦 ↔ 𝑣 βŠ† 𝑦))
113112rabbidv 3440 . . . . . . . . . . 11 (𝑧 = 𝑣 β†’ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})
114113cbvmptv 5261 . . . . . . . . . 10 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}) = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})
11514, 114eqtri 2760 . . . . . . . . 9 𝐹 = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦})
116 ineq2 4206 . . . . . . . . . . . 12 (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} β†’ (π‘Ž ∩ 𝑏) = (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}))
117116pweqd 4619 . . . . . . . . . . 11 (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦} β†’ 𝒫 (π‘Ž ∩ 𝑏) = 𝒫 (π‘Ž ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 βŠ† 𝑦}))
118117ineq2d 4212 . . . . . . . . . 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 3000 . . . . . . . . 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 7095 . . . . . . . 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 3177 . . . . . 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 256 . . . . 5 (𝐴 ∈ V β†’ βˆ€π‘Ž ∈ ran πΉβˆ€π‘ ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (π‘Ž ∩ 𝑏)) β‰  βˆ…)
12433, 49, 1233jca 1128 . . . 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 23340 . . . . 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 711 . . 3 (𝐴 ∈ V β†’ ran 𝐹 ∈ (fBasβ€˜π‘†))
1283, 127eqeltrid 2837 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆ‰ wnel 3046  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  Fincfn 8941  fBascfbas 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-en 8942  df-fin 8945  df-fbas 20947
This theorem is referenced by:  eltsms  23644  haustsms  23647  tsmscls  23649  tsmsmhm  23657  tsmsadd  23658
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