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Theorem pwcfsdom 9858
Description: A corollary of Konig's Theorem konigth 9844. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypothesis
Ref Expression
pwcfsdom.1 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))
Assertion
Ref Expression
pwcfsdom (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Distinct variable group:   𝐴,𝑓,𝑦
Allowed substitution hints:   𝐻(𝑦,𝑓)

Proof of Theorem pwcfsdom
Dummy variables 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onzsl 7424 . . . 4 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
21biimpi 217 . . 3 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3 cfom 9539 . . . . . . 7 (cf‘ω) = ω
4 aleph0 9345 . . . . . . . 8 (ℵ‘∅) = ω
54fveq2i 6548 . . . . . . 7 (cf‘(ℵ‘∅)) = (cf‘ω)
63, 5, 43eqtr4i 2831 . . . . . 6 (cf‘(ℵ‘∅)) = (ℵ‘∅)
7 2fveq3 6550 . . . . . 6 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
8 fveq2 6545 . . . . . 6 (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
96, 7, 83eqtr4a 2859 . . . . 5 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10 fvex 6558 . . . . . . . . 9 (ℵ‘𝐴) ∈ V
1110canth2 8524 . . . . . . . 8 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
1210pw2en 8478 . . . . . . . 8 𝒫 (ℵ‘𝐴) ≈ (2o𝑚 (ℵ‘𝐴))
13 sdomentr 8505 . . . . . . . 8 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2o𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2o𝑚 (ℵ‘𝐴)))
1411, 12, 13mp2an 688 . . . . . . 7 (ℵ‘𝐴) ≺ (2o𝑚 (ℵ‘𝐴))
15 alephon 9348 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
16 alephgeom 9361 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 omelon 8962 . . . . . . . . . . . 12 ω ∈ On
18 2onn 8123 . . . . . . . . . . . 12 2o ∈ ω
19 onelss 6115 . . . . . . . . . . . 12 (ω ∈ On → (2o ∈ ω → 2o ⊆ ω))
2017, 18, 19mp2 9 . . . . . . . . . . 11 2o ⊆ ω
21 sstr 3903 . . . . . . . . . . 11 ((2o ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2o ⊆ (ℵ‘𝐴))
2220, 21mpan 686 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → 2o ⊆ (ℵ‘𝐴))
2316, 22sylbi 218 . . . . . . . . 9 (𝐴 ∈ On → 2o ⊆ (ℵ‘𝐴))
24 ssdomg 8410 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (2o ⊆ (ℵ‘𝐴) → 2o ≼ (ℵ‘𝐴)))
2515, 23, 24mpsyl 68 . . . . . . . 8 (𝐴 ∈ On → 2o ≼ (ℵ‘𝐴))
26 mapdom1 8536 . . . . . . . 8 (2o ≼ (ℵ‘𝐴) → (2o𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
2725, 26syl 17 . . . . . . 7 (𝐴 ∈ On → (2o𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
28 sdomdomtr 8504 . . . . . . 7 (((ℵ‘𝐴) ≺ (2o𝑚 (ℵ‘𝐴)) ∧ (2o𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
2914, 27, 28sylancr 587 . . . . . 6 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
30 oveq2 7031 . . . . . . 7 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
3130breq2d 4980 . . . . . 6 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))))
3229, 31syl5ibrcom 248 . . . . 5 (𝐴 ∈ On → ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
339, 32syl5 34 . . . 4 (𝐴 ∈ On → (𝐴 = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
34 alephreg 9857 . . . . . . 7 (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35 2fveq3 6550 . . . . . . 7 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
36 fveq2 6545 . . . . . . 7 (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
3734, 35, 363eqtr4a 2859 . . . . . 6 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3837rexlimivw 3247 . . . . 5 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3938, 32syl5 34 . . . 4 (𝐴 ∈ On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
40 cfsmo 9546 . . . . . 6 ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)))
41 limelon 6136 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42 ffn 6389 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43 fnrnfv 6600 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
4443unieqd 4761 . . . . . . . . . . . . . . . 16 (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
4542, 44syl 17 . . . . . . . . . . . . . . 15 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
46 fvex 6558 . . . . . . . . . . . . . . . 16 (𝑓𝑥) ∈ V
4746dfiun2 4867 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)}
4845, 47syl6eqr 2851 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥))
4948ad2antrl 724 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ran 𝑓 = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥))
50 fnfvelrn 6720 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
5142, 50sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
52 sseq2 3920 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑓𝑤) → (𝑧𝑦𝑧 ⊆ (𝑓𝑤)))
5352rspcev 3561 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5451, 53sylan 580 . . . . . . . . . . . . . . . . . 18 (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5554rexlimdva2 3252 . . . . . . . . . . . . . . . . 17 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5655ralimdv 3147 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5756imp 407 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5857adantl 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦)
59 alephislim 9362 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
6059biimpi 217 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → Lim (ℵ‘𝐴))
61 frn 6395 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
6261adantr 481 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
63 coflim 9536 . . . . . . . . . . . . . . 15 ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → ( ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
6460, 62, 63syl2an 595 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ( ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
6558, 64mpbird 258 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ran 𝑓 = (ℵ‘𝐴))
6649, 65eqtr3d 2835 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = (ℵ‘𝐴))
67 ffvelrn 6721 . . . . . . . . . . . . . . . . 17 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ∈ (ℵ‘𝐴))
6815oneli 6180 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ (ℵ‘𝐴) → (𝑓𝑥) ∈ On)
69 harcard 9260 . . . . . . . . . . . . . . . . . . 19 (card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥))
70 iscard 9257 . . . . . . . . . . . . . . . . . . . 20 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) ↔ ((har‘(𝑓𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))))
7170simprbi 497 . . . . . . . . . . . . . . . . . . 19 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) → ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥)))
7269, 71ax-mp 5 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))
73 domrefg 8399 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ V → (𝑓𝑥) ≼ (𝑓𝑥))
7446, 73ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑥) ≼ (𝑓𝑥)
75 elharval 8880 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) ↔ ((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)))
7675biimpri 229 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)) → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
7774, 76mpan2 687 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ∈ On → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
78 breq1 4971 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑓𝑥) → (𝑦 ≺ (har‘(𝑓𝑥)) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
7978rspccv 3558 . . . . . . . . . . . . . . . . . 18 (∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥)) → ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) → (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8072, 77, 79mpsyl 68 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ On → (𝑓𝑥) ≺ (har‘(𝑓𝑥)))
8167, 68, 803syl 18 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (har‘(𝑓𝑥)))
82 harcl 8878 . . . . . . . . . . . . . . . . . . 19 (har‘(𝑓𝑥)) ∈ On
83 2fveq3 6550 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (har‘(𝑓𝑦)) = (har‘(𝑓𝑥)))
84 pwcfsdom.1 . . . . . . . . . . . . . . . . . . . 20 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))
8583, 84fvmptg 6640 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓𝑥)) ∈ On) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8682, 85mpan2 687 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8786breq2d 4980 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8887adantl 482 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8981, 88mpbird 258 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (𝐻𝑥))
9089ralrimiva 3151 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥))
91 fvex 6558 . . . . . . . . . . . . . . 15 (cf‘(ℵ‘𝐴)) ∈ V
92 eqid 2797 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥)
93 eqid 2797 . . . . . . . . . . . . . . 15 X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥)
9491, 92, 93konigth 9844 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9590, 94syl 17 . . . . . . . . . . . . 13 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9695ad2antrl 724 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9766, 96eqbrtrrd 4992 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9841, 97sylan 580 . . . . . . . . . 10 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
99 ovex 7055 . . . . . . . . . . . 12 ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V
10067ex 413 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓𝑥) ∈ (ℵ‘𝐴)))
101 alephlim 9346 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
102101eleq2d 2870 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦)))
103 eliun 4835 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦))
104 alephcard 9349 . . . . . . . . . . . . . . . . . . . . . . . 24 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
105104eleq2i 2876 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓𝑥) ∈ (ℵ‘𝑦))
106 cardsdomelir 9255 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓𝑥) ≺ (ℵ‘𝑦))
107105, 106sylbir 236 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (𝑓𝑥) ≺ (ℵ‘𝑦))
108 elharval 8880 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓𝑥)))
109108simprbi 497 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → (ℵ‘𝑦) ≼ (𝑓𝑥))
110 domnsym 8497 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℵ‘𝑦) ≼ (𝑓𝑥) → ¬ (𝑓𝑥) ≺ (ℵ‘𝑦))
111109, 110syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → ¬ (𝑓𝑥) ≺ (ℵ‘𝑦))
112111con2i 141 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
113 alephon 9348 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℵ‘𝑦) ∈ On
114 ontri1 6107 . . . . . . . . . . . . . . . . . . . . . . . 24 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥))))
11582, 113, 114mp2an 688 . . . . . . . . . . . . . . . . . . . . . . 23 ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
116112, 115sylibr 235 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
117107, 116syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
118 alephord2i 9356 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
119118imp 407 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
120 ontr2 6120 . . . . . . . . . . . . . . . . . . . . . 22 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12182, 15, 120mp2an 688 . . . . . . . . . . . . . . . . . . . . 21 (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
122117, 119, 121syl2anr 596 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑦𝐴) ∧ (𝑓𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
123122rexlimdva2 3252 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → (∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
124103, 123syl5bi 243 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12541, 124syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
126102, 125sylbid 241 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
127100, 126sylan9r 509 . . . . . . . . . . . . . . 15 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
128127imp 407 . . . . . . . . . . . . . 14 ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
12983cbvmptv 5068 . . . . . . . . . . . . . . 15 (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
13084, 129eqtri 2821 . . . . . . . . . . . . . 14 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
131128, 130fmptd 6748 . . . . . . . . . . . . 13 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
132 ffvelrn 6721 . . . . . . . . . . . . . . 15 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ∈ (ℵ‘𝐴))
133 onelss 6115 . . . . . . . . . . . . . . 15 ((ℵ‘𝐴) ∈ On → ((𝐻𝑥) ∈ (ℵ‘𝐴) → (𝐻𝑥) ⊆ (ℵ‘𝐴)))
13415, 132, 133mpsyl 68 . . . . . . . . . . . . . 14 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ⊆ (ℵ‘𝐴))
135134ralrimiva 3151 . . . . . . . . . . . . 13 (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴))
136 ss2ixp 8330 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
13791, 10ixpconst 8327 . . . . . . . . . . . . . 14 X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
138136, 137syl6sseq 3944 . . . . . . . . . . . . 13 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
139131, 135, 1383syl 18 . . . . . . . . . . . 12 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
140 ssdomg 8410 . . . . . . . . . . . 12 (((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V → (X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
14199, 139, 140mpsyl 68 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
142141adantrr 713 . . . . . . . . . 10 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
143 sdomdomtr 8504 . . . . . . . . . 10 (((ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ∧ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
14498, 142, 143syl2anc 584 . . . . . . . . 9 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
145144expcom 414 . . . . . . . 8 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
1461453adant2 1124 . . . . . . 7 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
147146exlimiv 1912 . . . . . 6 (∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
14815, 40, 147mp2b 10 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
149148a1i 11 . . . 4 (𝐴 ∈ On → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
15033, 39, 1493jaod 1421 . . 3 (𝐴 ∈ On → ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
1512, 150mpd 15 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
152 alephfnon 9344 . . . . 5 ℵ Fn On
153 fndm 6332 . . . . 5 (ℵ Fn On → dom ℵ = On)
154152, 153ax-mp 5 . . . 4 dom ℵ = On
155154eleq2i 2876 . . 3 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
156 ndmfv 6575 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
157 1n0 7977 . . . . . 6 1o ≠ ∅
158 1oex 7968 . . . . . . 7 1o ∈ V
1591580sdom 8502 . . . . . 6 (∅ ≺ 1o ↔ 1o ≠ ∅)
160157, 159mpbir 232 . . . . 5 ∅ ≺ 1o
161 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
162 fveq2 6545 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
163 cf0 9526 . . . . . . . . 9 (cf‘∅) = ∅
164162, 163syl6eq 2849 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
165161, 164oveq12d 7041 . . . . . . 7 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = (∅ ↑𝑚 ∅))
166 0ex 5109 . . . . . . . 8 ∅ ∈ V
167 map0e 8303 . . . . . . . 8 (∅ ∈ V → (∅ ↑𝑚 ∅) = 1o)
168166, 167ax-mp 5 . . . . . . 7 (∅ ↑𝑚 ∅) = 1o
169165, 168syl6eq 2849 . . . . . 6 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = 1o)
170161, 169breq12d 4981 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1o))
171160, 170mpbiri 259 . . . 4 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
172156, 171syl 17 . . 3 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
173155, 172sylnbir 332 . 2 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
174151, 173pm2.61i 183 1 (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3o 1079  w3a 1080   = wceq 1525  wex 1765  wcel 2083  {cab 2777  wne 2986  wral 3107  wrex 3108  Vcvv 3440  wss 3865  c0 4217  𝒫 cpw 4459   cuni 4751   ciun 4831   class class class wbr 4968  cmpt 5047  dom cdm 5450  ran crn 5451  Oncon0 6073  Lim wlim 6074  suc csuc 6075   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  ωcom 7443  Smo wsmo 7841  1oc1o 7953  2oc2o 7954  𝑚 cmap 8263  Xcixp 8317  cen 8361  cdom 8362  csdm 8363  harchar 8873  cardccrd 9217  cale 9218  cfccf 9219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-ac2 9738
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-smo 7842  df-recs 7867  df-rdg 7905  df-1o 7960  df-2o 7961  df-oadd 7964  df-er 8146  df-map 8265  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-oi 8827  df-har 8875  df-card 9221  df-aleph 9222  df-cf 9223  df-acn 9224  df-ac 9395
This theorem is referenced by:  cfpwsdom  9859  tskcard  10056  bj-pwcfsdom  33974
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