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Theorem tsmsfbas 22670
 Description: The collection of all sets of the form 𝐹(𝑧) = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 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 = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
tsmsfbas.l 𝐿 = ran 𝐹
tsmsfbas.a (𝜑𝐴𝑊)
Assertion
Ref Expression
tsmsfbas (𝜑𝐿 ∈ (fBas‘𝑆))
Distinct variable groups:   𝑧,𝐴   𝑦,𝑧,𝑆
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑦)   𝐹(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem tsmsfbas
Dummy variables 𝑢 𝑎 𝑣 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmsfbas.a . 2 (𝜑𝐴𝑊)
2 elex 3518 . 2 (𝐴𝑊𝐴 ∈ V)
3 tsmsfbas.l . . 3 𝐿 = ran 𝐹
4 ssrab2 4060 . . . . . . 7 {𝑦𝑆𝑧𝑦} ⊆ 𝑆
5 tsmsfbas.s . . . . . . . . . 10 𝑆 = (𝒫 𝐴 ∩ Fin)
6 pwexg 5276 . . . . . . . . . . 11 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
7 inex1g 5220 . . . . . . . . . . 11 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
86, 7syl 17 . . . . . . . . . 10 (𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
95, 8eqeltrid 2922 . . . . . . . . 9 (𝐴 ∈ V → 𝑆 ∈ V)
109adantr 481 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑧𝑆) → 𝑆 ∈ V)
11 elpw2g 5244 . . . . . . . 8 (𝑆 ∈ V → ({𝑦𝑆𝑧𝑦} ∈ 𝒫 𝑆 ↔ {𝑦𝑆𝑧𝑦} ⊆ 𝑆))
1210, 11syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ 𝑧𝑆) → ({𝑦𝑆𝑧𝑦} ∈ 𝒫 𝑆 ↔ {𝑦𝑆𝑧𝑦} ⊆ 𝑆))
134, 12mpbiri 259 . . . . . 6 ((𝐴 ∈ V ∧ 𝑧𝑆) → {𝑦𝑆𝑧𝑦} ∈ 𝒫 𝑆)
14 tsmsfbas.f . . . . . 6 𝐹 = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
1513, 14fmptd 6876 . . . . 5 (𝐴 ∈ V → 𝐹:𝑆⟶𝒫 𝑆)
1615frnd 6520 . . . 4 (𝐴 ∈ V → ran 𝐹 ⊆ 𝒫 𝑆)
17 0ss 4354 . . . . . . . . . 10 ∅ ⊆ 𝐴
18 0fin 8740 . . . . . . . . . 10 ∅ ∈ Fin
19 elfpw 8820 . . . . . . . . . 10 (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin))
2017, 18, 19mpbir2an 707 . . . . . . . . 9 ∅ ∈ (𝒫 𝐴 ∩ Fin)
2120, 5eleqtrri 2917 . . . . . . . 8 ∅ ∈ 𝑆
22 0ss 4354 . . . . . . . . 9 ∅ ⊆ 𝑦
2322rgenw 3155 . . . . . . . 8 𝑦𝑆 ∅ ⊆ 𝑦
24 rabid2 3387 . . . . . . . . . 10 (𝑆 = {𝑦𝑆𝑧𝑦} ↔ ∀𝑦𝑆 𝑧𝑦)
25 sseq1 3996 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑧𝑦 ↔ ∅ ⊆ 𝑦))
2625ralbidv 3202 . . . . . . . . . 10 (𝑧 = ∅ → (∀𝑦𝑆 𝑧𝑦 ↔ ∀𝑦𝑆 ∅ ⊆ 𝑦))
2724, 26syl5bb 284 . . . . . . . . 9 (𝑧 = ∅ → (𝑆 = {𝑦𝑆𝑧𝑦} ↔ ∀𝑦𝑆 ∅ ⊆ 𝑦))
2827rspcev 3627 . . . . . . . 8 ((∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ∅ ⊆ 𝑦) → ∃𝑧𝑆 𝑆 = {𝑦𝑆𝑧𝑦})
2921, 23, 28mp2an 688 . . . . . . 7 𝑧𝑆 𝑆 = {𝑦𝑆𝑧𝑦}
3014elrnmpt 5827 . . . . . . . 8 (𝑆 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧𝑆 𝑆 = {𝑦𝑆𝑧𝑦}))
319, 30syl 17 . . . . . . 7 (𝐴 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧𝑆 𝑆 = {𝑦𝑆𝑧𝑦}))
3229, 31mpbiri 259 . . . . . 6 (𝐴 ∈ V → 𝑆 ∈ ran 𝐹)
3332ne0d 4305 . . . . 5 (𝐴 ∈ V → ran 𝐹 ≠ ∅)
34 simpr 485 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝑧𝑆) → 𝑧𝑆)
35 ssid 3993 . . . . . . . . . . . 12 𝑧𝑧
36 sseq2 3997 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → (𝑧𝑦𝑧𝑧))
3736rspcev 3627 . . . . . . . . . . . 12 ((𝑧𝑆𝑧𝑧) → ∃𝑦𝑆 𝑧𝑦)
3834, 35, 37sylancl 586 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑧𝑆) → ∃𝑦𝑆 𝑧𝑦)
39 rabn0 4343 . . . . . . . . . . 11 ({𝑦𝑆𝑧𝑦} ≠ ∅ ↔ ∃𝑦𝑆 𝑧𝑦)
4038, 39sylibr 235 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑧𝑆) → {𝑦𝑆𝑧𝑦} ≠ ∅)
4140necomd 3076 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑧𝑆) → ∅ ≠ {𝑦𝑆𝑧𝑦})
4241neneqd 3026 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑧𝑆) → ¬ ∅ = {𝑦𝑆𝑧𝑦})
4342nrexdv 3275 . . . . . . 7 (𝐴 ∈ V → ¬ ∃𝑧𝑆 ∅ = {𝑦𝑆𝑧𝑦})
44 0ex 5208 . . . . . . . 8 ∅ ∈ V
4514elrnmpt 5827 . . . . . . . 8 (∅ ∈ V → (∅ ∈ ran 𝐹 ↔ ∃𝑧𝑆 ∅ = {𝑦𝑆𝑧𝑦}))
4644, 45ax-mp 5 . . . . . . 7 (∅ ∈ ran 𝐹 ↔ ∃𝑧𝑆 ∅ = {𝑦𝑆𝑧𝑦})
4743, 46sylnibr 330 . . . . . 6 (𝐴 ∈ V → ¬ ∅ ∈ ran 𝐹)
48 df-nel 3129 . . . . . 6 (∅ ∉ ran 𝐹 ↔ ¬ ∅ ∈ ran 𝐹)
4947, 48sylibr 235 . . . . 5 (𝐴 ∈ V → ∅ ∉ ran 𝐹)
50 elfpw 8820 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑢𝐴𝑢 ∈ Fin))
5150simplbi 498 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢𝐴)
5251, 5eleq2s 2936 . . . . . . . . . . . . . . . 16 (𝑢𝑆𝑢𝐴)
53 elfpw 8820 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣𝐴𝑣 ∈ Fin))
5453simplbi 498 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣𝐴)
5554, 5eleq2s 2936 . . . . . . . . . . . . . . . 16 (𝑣𝑆𝑣𝐴)
5652, 55anim12i 612 . . . . . . . . . . . . . . 15 ((𝑢𝑆𝑣𝑆) → (𝑢𝐴𝑣𝐴))
57 unss 4164 . . . . . . . . . . . . . . 15 ((𝑢𝐴𝑣𝐴) ↔ (𝑢𝑣) ⊆ 𝐴)
5856, 57sylib 219 . . . . . . . . . . . . . 14 ((𝑢𝑆𝑣𝑆) → (𝑢𝑣) ⊆ 𝐴)
59 elinel2 4177 . . . . . . . . . . . . . . . 16 (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ∈ Fin)
6059, 5eleq2s 2936 . . . . . . . . . . . . . . 15 (𝑢𝑆𝑢 ∈ Fin)
61 elinel2 4177 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ∈ Fin)
6261, 5eleq2s 2936 . . . . . . . . . . . . . . 15 (𝑣𝑆𝑣 ∈ Fin)
63 unfi 8779 . . . . . . . . . . . . . . 15 ((𝑢 ∈ Fin ∧ 𝑣 ∈ Fin) → (𝑢𝑣) ∈ Fin)
6460, 62, 63syl2an 595 . . . . . . . . . . . . . 14 ((𝑢𝑆𝑣𝑆) → (𝑢𝑣) ∈ Fin)
65 elfpw 8820 . . . . . . . . . . . . . 14 ((𝑢𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑢𝑣) ⊆ 𝐴 ∧ (𝑢𝑣) ∈ Fin))
6658, 64, 65sylanbrc 583 . . . . . . . . . . . . 13 ((𝑢𝑆𝑣𝑆) → (𝑢𝑣) ∈ (𝒫 𝐴 ∩ Fin))
6766adantl 482 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝑣) ∈ (𝒫 𝐴 ∩ Fin))
6867, 5syl6eleqr 2929 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝑣) ∈ 𝑆)
69 eqidd 2827 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} = {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦})
70 sseq1 3996 . . . . . . . . . . . . 13 (𝑎 = (𝑢𝑣) → (𝑎𝑦 ↔ (𝑢𝑣) ⊆ 𝑦))
7170rabbidv 3486 . . . . . . . . . . . 12 (𝑎 = (𝑢𝑣) → {𝑦𝑆𝑎𝑦} = {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦})
7271rspceeqv 3642 . . . . . . . . . . 11 (((𝑢𝑣) ∈ 𝑆 ∧ {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} = {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) → ∃𝑎𝑆 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} = {𝑦𝑆𝑎𝑦})
7368, 69, 72syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → ∃𝑎𝑆 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} = {𝑦𝑆𝑎𝑦})
749adantr 481 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → 𝑆 ∈ V)
75 rabexg 5231 . . . . . . . . . . . 12 (𝑆 ∈ V → {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ V)
7674, 75syl 17 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ V)
77 sseq1 3996 . . . . . . . . . . . . . . 15 (𝑧 = 𝑎 → (𝑧𝑦𝑎𝑦))
7877rabbidv 3486 . . . . . . . . . . . . . 14 (𝑧 = 𝑎 → {𝑦𝑆𝑧𝑦} = {𝑦𝑆𝑎𝑦})
7978cbvmptv 5166 . . . . . . . . . . . . 13 (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}) = (𝑎𝑆 ↦ {𝑦𝑆𝑎𝑦})
8014, 79eqtri 2849 . . . . . . . . . . . 12 𝐹 = (𝑎𝑆 ↦ {𝑦𝑆𝑎𝑦})
8180elrnmpt 5827 . . . . . . . . . . 11 ({𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ V → ({𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ ran 𝐹 ↔ ∃𝑎𝑆 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} = {𝑦𝑆𝑎𝑦}))
8276, 81syl 17 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → ({𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ ran 𝐹 ↔ ∃𝑎𝑆 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} = {𝑦𝑆𝑎𝑦}))
8373, 82mpbird 258 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ ran 𝐹)
84 pwidg 4559 . . . . . . . . . 10 ({𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ V → {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦})
8576, 84syl 17 . . . . . . . . 9 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦})
86 inelcm 4417 . . . . . . . . 9 (({𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ ran 𝐹 ∧ {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) → (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅)
8783, 85, 86syl2anc 584 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝑢𝑆𝑣𝑆)) → (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅)
8887ralrimivva 3196 . . . . . . 7 (𝐴 ∈ V → ∀𝑢𝑆𝑣𝑆 (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅)
89 rabexg 5231 . . . . . . . . . 10 (𝑆 ∈ V → {𝑦𝑆𝑢𝑦} ∈ V)
909, 89syl 17 . . . . . . . . 9 (𝐴 ∈ V → {𝑦𝑆𝑢𝑦} ∈ V)
9190ralrimivw 3188 . . . . . . . 8 (𝐴 ∈ V → ∀𝑢𝑆 {𝑦𝑆𝑢𝑦} ∈ V)
92 sseq1 3996 . . . . . . . . . . . 12 (𝑧 = 𝑢 → (𝑧𝑦𝑢𝑦))
9392rabbidv 3486 . . . . . . . . . . 11 (𝑧 = 𝑢 → {𝑦𝑆𝑧𝑦} = {𝑦𝑆𝑢𝑦})
9493cbvmptv 5166 . . . . . . . . . 10 (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}) = (𝑢𝑆 ↦ {𝑦𝑆𝑢𝑦})
9514, 94eqtri 2849 . . . . . . . . 9 𝐹 = (𝑢𝑆 ↦ {𝑦𝑆𝑢𝑦})
96 ineq1 4185 . . . . . . . . . . . . . 14 (𝑎 = {𝑦𝑆𝑢𝑦} → (𝑎 ∩ {𝑦𝑆𝑣𝑦}) = ({𝑦𝑆𝑢𝑦} ∩ {𝑦𝑆𝑣𝑦}))
97 inrab 4279 . . . . . . . . . . . . . . 15 ({𝑦𝑆𝑢𝑦} ∩ {𝑦𝑆𝑣𝑦}) = {𝑦𝑆 ∣ (𝑢𝑦𝑣𝑦)}
98 unss 4164 . . . . . . . . . . . . . . . 16 ((𝑢𝑦𝑣𝑦) ↔ (𝑢𝑣) ⊆ 𝑦)
9998rabbii 3479 . . . . . . . . . . . . . . 15 {𝑦𝑆 ∣ (𝑢𝑦𝑣𝑦)} = {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}
10097, 99eqtri 2849 . . . . . . . . . . . . . 14 ({𝑦𝑆𝑢𝑦} ∩ {𝑦𝑆𝑣𝑦}) = {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}
10196, 100syl6eq 2877 . . . . . . . . . . . . 13 (𝑎 = {𝑦𝑆𝑢𝑦} → (𝑎 ∩ {𝑦𝑆𝑣𝑦}) = {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦})
102101pweqd 4547 . . . . . . . . . . . 12 (𝑎 = {𝑦𝑆𝑢𝑦} → 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦}) = 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦})
103102ineq2d 4193 . . . . . . . . . . 11 (𝑎 = {𝑦𝑆𝑢𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) = (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}))
104103neeq1d 3080 . . . . . . . . . 10 (𝑎 = {𝑦𝑆𝑢𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅))
105104ralbidv 3202 . . . . . . . . 9 (𝑎 = {𝑦𝑆𝑢𝑦} → (∀𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅ ↔ ∀𝑣𝑆 (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅))
10695, 105ralrnmptw 6858 . . . . . . . 8 (∀𝑢𝑆 {𝑦𝑆𝑢𝑦} ∈ V → (∀𝑎 ∈ ran 𝐹𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅ ↔ ∀𝑢𝑆𝑣𝑆 (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅))
10791, 106syl 17 . . . . . . 7 (𝐴 ∈ V → (∀𝑎 ∈ ran 𝐹𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅ ↔ ∀𝑢𝑆𝑣𝑆 (ran 𝐹 ∩ 𝒫 {𝑦𝑆 ∣ (𝑢𝑣) ⊆ 𝑦}) ≠ ∅))
10888, 107mpbird 258 . . . . . 6 (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅)
109 rabexg 5231 . . . . . . . . . 10 (𝑆 ∈ V → {𝑦𝑆𝑣𝑦} ∈ V)
1109, 109syl 17 . . . . . . . . 9 (𝐴 ∈ V → {𝑦𝑆𝑣𝑦} ∈ V)
111110ralrimivw 3188 . . . . . . . 8 (𝐴 ∈ V → ∀𝑣𝑆 {𝑦𝑆𝑣𝑦} ∈ V)
112 sseq1 3996 . . . . . . . . . . . 12 (𝑧 = 𝑣 → (𝑧𝑦𝑣𝑦))
113112rabbidv 3486 . . . . . . . . . . 11 (𝑧 = 𝑣 → {𝑦𝑆𝑧𝑦} = {𝑦𝑆𝑣𝑦})
114113cbvmptv 5166 . . . . . . . . . 10 (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}) = (𝑣𝑆 ↦ {𝑦𝑆𝑣𝑦})
11514, 114eqtri 2849 . . . . . . . . 9 𝐹 = (𝑣𝑆 ↦ {𝑦𝑆𝑣𝑦})
116 ineq2 4187 . . . . . . . . . . . 12 (𝑏 = {𝑦𝑆𝑣𝑦} → (𝑎𝑏) = (𝑎 ∩ {𝑦𝑆𝑣𝑦}))
117116pweqd 4547 . . . . . . . . . . 11 (𝑏 = {𝑦𝑆𝑣𝑦} → 𝒫 (𝑎𝑏) = 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦}))
118117ineq2d 4193 . . . . . . . . . 10 (𝑏 = {𝑦𝑆𝑣𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎𝑏)) = (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})))
119118neeq1d 3080 . . . . . . . . 9 (𝑏 = {𝑦𝑆𝑣𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅))
120115, 119ralrnmptw 6858 . . . . . . . 8 (∀𝑣𝑆 {𝑦𝑆𝑣𝑦} ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅ ↔ ∀𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅))
121111, 120syl 17 . . . . . . 7 (𝐴 ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅ ↔ ∀𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅))
122121ralbidv 3202 . . . . . 6 (𝐴 ∈ V → (∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅ ↔ ∀𝑎 ∈ ran 𝐹𝑣𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦𝑆𝑣𝑦})) ≠ ∅))
123108, 122mpbird 258 . . . . 5 (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅)
12433, 49, 1233jca 1122 . . . 4 (𝐴 ∈ V → (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅))
125 isfbas 22372 . . . . 5 (𝑆 ∈ V → (ran 𝐹 ∈ (fBas‘𝑆) ↔ (ran 𝐹 ⊆ 𝒫 𝑆 ∧ (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅))))
1269, 125syl 17 . . . 4 (𝐴 ∈ V → (ran 𝐹 ∈ (fBas‘𝑆) ↔ (ran 𝐹 ⊆ 𝒫 𝑆 ∧ (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran 𝐹 ∧ ∀𝑎 ∈ ran 𝐹𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎𝑏)) ≠ ∅))))
12716, 124, 126mpbir2and 709 . . 3 (𝐴 ∈ V → ran 𝐹 ∈ (fBas‘𝑆))
1283, 127eqeltrid 2922 . 2 (𝐴 ∈ V → 𝐿 ∈ (fBas‘𝑆))
1291, 2, 1283syl 18 1 (𝜑𝐿 ∈ (fBas‘𝑆))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3021   ∉ wnel 3128  ∀wral 3143  ∃wrex 3144  {crab 3147  Vcvv 3500   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295  𝒫 cpw 4542   ↦ cmpt 5143  ran crn 5555  ‘cfv 6354  Fincfn 8503  fBascfbas 20468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102  df-er 8284  df-en 8504  df-fin 8507  df-fbas 20477 This theorem is referenced by:  eltsms  22675  haustsms  22678  tsmscls  22680  tsmsmhm  22688  tsmsadd  22689
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