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Theorem pwcfsdom 9998
 Description: A corollary of Konig's Theorem konigth 9984. 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 7545 . . . 4 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
21biimpi 219 . . 3 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3 cfom 9679 . . . . . . 7 (cf‘ω) = ω
4 aleph0 9481 . . . . . . . 8 (ℵ‘∅) = ω
54fveq2i 6652 . . . . . . 7 (cf‘(ℵ‘∅)) = (cf‘ω)
63, 5, 43eqtr4i 2834 . . . . . 6 (cf‘(ℵ‘∅)) = (ℵ‘∅)
7 2fveq3 6654 . . . . . 6 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
8 fveq2 6649 . . . . . 6 (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
96, 7, 83eqtr4a 2862 . . . . 5 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10 fvex 6662 . . . . . . . . 9 (ℵ‘𝐴) ∈ V
1110canth2 8658 . . . . . . . 8 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
1210pw2en 8611 . . . . . . . 8 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))
13 sdomentr 8639 . . . . . . . 8 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2om (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)))
1411, 12, 13mp2an 691 . . . . . . 7 (ℵ‘𝐴) ≺ (2om (ℵ‘𝐴))
15 alephon 9484 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
16 alephgeom 9497 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 omelon 9097 . . . . . . . . . . . 12 ω ∈ On
18 2onn 8253 . . . . . . . . . . . 12 2o ∈ ω
19 onelss 6205 . . . . . . . . . . . 12 (ω ∈ On → (2o ∈ ω → 2o ⊆ ω))
2017, 18, 19mp2 9 . . . . . . . . . . 11 2o ⊆ ω
21 sstr 3926 . . . . . . . . . . 11 ((2o ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2o ⊆ (ℵ‘𝐴))
2220, 21mpan 689 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → 2o ⊆ (ℵ‘𝐴))
2316, 22sylbi 220 . . . . . . . . 9 (𝐴 ∈ On → 2o ⊆ (ℵ‘𝐴))
24 ssdomg 8542 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (2o ⊆ (ℵ‘𝐴) → 2o ≼ (ℵ‘𝐴)))
2515, 23, 24mpsyl 68 . . . . . . . 8 (𝐴 ∈ On → 2o ≼ (ℵ‘𝐴))
26 mapdom1 8670 . . . . . . . 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 8638 . . . . . . 7 (((ℵ‘𝐴) ≺ (2om (ℵ‘𝐴)) ∧ (2om (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (ℵ‘𝐴)))
2914, 27, 28sylancr 590 . . . . . 6 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (ℵ‘𝐴)))
30 oveq2 7147 . . . . . . 7 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑m (ℵ‘𝐴)))
3130breq2d 5045 . . . . . 6 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) ↔ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (ℵ‘𝐴))))
3229, 31syl5ibrcom 250 . . . . 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 9997 . . . . . . 7 (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35 2fveq3 6654 . . . . . . 7 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
36 fveq2 6649 . . . . . . 7 (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
3734, 35, 363eqtr4a 2862 . . . . . 6 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3837rexlimivw 3244 . . . . 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 9686 . . . . . 6 ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)))
41 limelon 6226 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42 ffn 6491 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43 fnrnfv 6704 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
4443unieqd 4817 . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . 16 (𝑓𝑥) ∈ V
4746dfiun2 4923 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)}
4845, 47eqtr4di 2854 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥))
4948ad2antrl 727 . . . . . . . . . . . . 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 6829 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
5142, 50sylan 583 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
52 sseq2 3944 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑓𝑤) → (𝑧𝑦𝑧 ⊆ (𝑓𝑤)))
5352rspcev 3574 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5451, 53sylan 583 . . . . . . . . . . . . . . . . . 18 (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5554rexlimdva2 3249 . . . . . . . . . . . . . . . . 17 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5655ralimdv 3148 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5756imp 410 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5857adantl 485 . . . . . . . . . . . . . 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 9498 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
6059biimpi 219 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → Lim (ℵ‘𝐴))
61 frn 6497 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
6261adantr 484 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
63 coflim 9676 . . . . . . . . . . . . . . 15 ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → ( ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
6460, 62, 63syl2an 598 . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ran 𝑓 = (ℵ‘𝐴))
6649, 65eqtr3d 2838 . . . . . . . . . . . 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 6830 . . . . . . . . . . . . . . . . 17 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ∈ (ℵ‘𝐴))
6815oneli 6270 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ (ℵ‘𝐴) → (𝑓𝑥) ∈ On)
69 harcard 9395 . . . . . . . . . . . . . . . . . . 19 (card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥))
70 iscard 9392 . . . . . . . . . . . . . . . . . . . 20 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) ↔ ((har‘(𝑓𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))))
7170simprbi 500 . . . . . . . . . . . . . . . . . . 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 8531 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ V → (𝑓𝑥) ≼ (𝑓𝑥))
7446, 73ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑥) ≼ (𝑓𝑥)
75 elharval 9013 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) ↔ ((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)))
7675biimpri 231 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)) → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
7774, 76mpan2 690 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ∈ On → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
78 breq1 5036 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑓𝑥) → (𝑦 ≺ (har‘(𝑓𝑥)) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
7978rspccv 3571 . . . . . . . . . . . . . . . . . 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 9011 . . . . . . . . . . . . . . . . . . 19 (har‘(𝑓𝑥)) ∈ On
83 2fveq3 6654 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (har‘(𝑓𝑦)) = (har‘(𝑓𝑥)))
84 pwcfsdom.1 . . . . . . . . . . . . . . . . . . . 20 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))
8583, 84fvmptg 6747 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓𝑥)) ∈ On) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8682, 85mpan2 690 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8786breq2d 5045 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8887adantl 485 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8981, 88mpbird 260 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (𝐻𝑥))
9089ralrimiva 3152 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥))
91 fvex 6662 . . . . . . . . . . . . . . 15 (cf‘(ℵ‘𝐴)) ∈ V
92 eqid 2801 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥)
93 eqid 2801 . . . . . . . . . . . . . . 15 X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥)
9491, 92, 93konigth 9984 . . . . . . . . . . . . . 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 727 . . . . . . . . . . . 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 5057 . . . . . . . . . . 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 583 . . . . . . . . . 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 7172 . . . . . . . . . . . 12 ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) ∈ V
10067ex 416 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓𝑥) ∈ (ℵ‘𝐴)))
101 alephlim 9482 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
102101eleq2d 2878 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦)))
103 eliun 4888 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦))
104 alephcard 9485 . . . . . . . . . . . . . . . . . . . . . . . 24 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
105104eleq2i 2884 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓𝑥) ∈ (ℵ‘𝑦))
106 cardsdomelir 9390 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓𝑥) ≺ (ℵ‘𝑦))
107105, 106sylbir 238 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (𝑓𝑥) ≺ (ℵ‘𝑦))
108 elharval 9013 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓𝑥)))
109108simprbi 500 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → (ℵ‘𝑦) ≼ (𝑓𝑥))
110 domnsym 8631 . . . . . . . . . . . . . . . . . . . . . . . . 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 9484 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℵ‘𝑦) ∈ On
114 ontri1 6197 . . . . . . . . . . . . . . . . . . . . . . . 24 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥))))
11582, 113, 114mp2an 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
116112, 115sylibr 237 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
117107, 116syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
118 alephord2i 9492 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
119118imp 410 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
120 ontr2 6210 . . . . . . . . . . . . . . . . . . . . . 22 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12182, 15, 120mp2an 691 . . . . . . . . . . . . . . . . . . . . 21 (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
122117, 119, 121syl2anr 599 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑦𝐴) ∧ (𝑓𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
123122rexlimdva2 3249 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → (∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
124103, 123syl5bi 245 . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
127100, 126sylan9r 512 . . . . . . . . . . . . . . 15 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
128127imp 410 . . . . . . . . . . . . . 14 ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
12983cbvmptv 5136 . . . . . . . . . . . . . . 15 (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
13084, 129eqtri 2824 . . . . . . . . . . . . . 14 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
131128, 130fmptd 6859 . . . . . . . . . . . . 13 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
132 ffvelrn 6830 . . . . . . . . . . . . . . 15 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ∈ (ℵ‘𝐴))
133 onelss 6205 . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . 13 (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴))
136 ss2ixp 8461 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
13791, 10ixpconst 8458 . . . . . . . . . . . . . 14 X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
138136, 137sseqtrdi 3968 . . . . . . . . . . . . 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 8542 . . . . . . . . . . . 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 716 . . . . . . . . . 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 8638 . . . . . . . . . 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 587 . . . . . . . . 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 417 . . . . . . . 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 1128 . . . . . . 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 1931 . . . . . 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 1425 . . 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 9480 . . . . 5 ℵ Fn On
153152fndmi 6430 . . . 4 dom ℵ = On
154153eleq2i 2884 . . 3 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
155 ndmfv 6679 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
156 1n0 8106 . . . . . 6 1o ≠ ∅
157 1oex 8097 . . . . . . 7 1o ∈ V
1581570sdom 8636 . . . . . 6 (∅ ≺ 1o ↔ 1o ≠ ∅)
159156, 158mpbir 234 . . . . 5 ∅ ≺ 1o
160 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
161 fveq2 6649 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
162 cf0 9666 . . . . . . . . 9 (cf‘∅) = ∅
163161, 162eqtrdi 2852 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
164160, 163oveq12d 7157 . . . . . . 7 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) = (∅ ↑m ∅))
165 0ex 5178 . . . . . . . 8 ∅ ∈ V
166 map0e 8433 . . . . . . . 8 (∅ ∈ V → (∅ ↑m ∅) = 1o)
167165, 166ax-mp 5 . . . . . . 7 (∅ ↑m ∅) = 1o
168164, 167eqtrdi 2852 . . . . . 6 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) = 1o)
169160, 168breq12d 5046 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1o))
170159, 169mpbiri 261 . . . 4 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))))
171155, 170syl 17 . . 3 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))))
172154, 171sylnbir 334 . 2 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))))
173151, 172pm2.61i 185 1 (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112  {cab 2779   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ⊆ wss 3884  ∅c0 4246  𝒫 cpw 4500  ∪ cuni 4803  ∪ ciun 4884   class class class wbr 5033   ↦ cmpt 5113  dom cdm 5523  ran crn 5524  Oncon0 6163  Lim wlim 6164  suc csuc 6165   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  ωcom 7564  Smo wsmo 7969  1oc1o 8082  2oc2o 8083   ↑m cmap 8393  Xcixp 8448   ≈ cen 8493   ≼ cdom 8494   ≺ csdm 8495  harchar 9008  cardccrd 9352  ℵcale 9353  cfccf 9354 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-ac2 9878 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-smo 7970  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-har 9009  df-card 9356  df-aleph 9357  df-cf 9358  df-acn 9359  df-ac 9531 This theorem is referenced by:  cfpwsdom  9999  tskcard  10196  bj-pwcfsdom  34474
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