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Theorem pwcfsdom 9390
Description: A corollary of Konig's Theorem konigth 9376. 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 7031 . . . 4 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
21biimpi 206 . . 3 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3 cfom 9071 . . . . . . 7 (cf‘ω) = ω
4 aleph0 8874 . . . . . . . 8 (ℵ‘∅) = ω
54fveq2i 6181 . . . . . . 7 (cf‘(ℵ‘∅)) = (cf‘ω)
63, 5, 43eqtr4i 2652 . . . . . 6 (cf‘(ℵ‘∅)) = (ℵ‘∅)
7 fveq2 6178 . . . . . . 7 (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
87fveq2d 6182 . . . . . 6 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
96, 8, 73eqtr4a 2680 . . . . 5 (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10 fvex 6188 . . . . . . . . 9 (ℵ‘𝐴) ∈ V
1110canth2 8098 . . . . . . . 8 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
1210pw2en 8052 . . . . . . . 8 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))
13 sdomentr 8079 . . . . . . . 8 (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)))
1411, 12, 13mp2an 707 . . . . . . 7 (ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴))
15 alephon 8877 . . . . . . . . 9 (ℵ‘𝐴) ∈ On
16 alephgeom 8890 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17 omelon 8528 . . . . . . . . . . . 12 ω ∈ On
18 2onn 7705 . . . . . . . . . . . 12 2𝑜 ∈ ω
19 onelss 5754 . . . . . . . . . . . 12 (ω ∈ On → (2𝑜 ∈ ω → 2𝑜 ⊆ ω))
2017, 18, 19mp2 9 . . . . . . . . . . 11 2𝑜 ⊆ ω
21 sstr 3603 . . . . . . . . . . 11 ((2𝑜 ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2𝑜 ⊆ (ℵ‘𝐴))
2220, 21mpan 705 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → 2𝑜 ⊆ (ℵ‘𝐴))
2316, 22sylbi 207 . . . . . . . . 9 (𝐴 ∈ On → 2𝑜 ⊆ (ℵ‘𝐴))
24 ssdomg 7986 . . . . . . . . 9 ((ℵ‘𝐴) ∈ On → (2𝑜 ⊆ (ℵ‘𝐴) → 2𝑜 ≼ (ℵ‘𝐴)))
2515, 23, 24mpsyl 68 . . . . . . . 8 (𝐴 ∈ On → 2𝑜 ≼ (ℵ‘𝐴))
26 mapdom1 8110 . . . . . . . 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 8078 . . . . . . 7 (((ℵ‘𝐴) ≺ (2𝑜𝑚 (ℵ‘𝐴)) ∧ (2𝑜𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
2914, 27, 28sylancr 694 . . . . . 6 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
30 oveq2 6643 . . . . . . 7 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
3130breq2d 4656 . . . . . 6 ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))))
3229, 31syl5ibrcom 237 . . . . 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 9389 . . . . . . 7 (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35 fveq2 6178 . . . . . . . 8 (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
3635fveq2d 6182 . . . . . . 7 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
3734, 36, 353eqtr4a 2680 . . . . . 6 (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
3837rexlimivw 3025 . . . . 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 9078 . . . . . 6 ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)))
41 limelon 5776 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42 ffn 6032 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43 fnrnfv 6229 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)})
4443unieqd 4437 . . . . . . . . . . . . . . . 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 6188 . . . . . . . . . . . . . . . 16 (𝑓𝑥) ∈ V
4746dfiun2 4545 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓𝑥)}
4845, 47syl6eqr 2672 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥))
4948ad2antrl 763 . . . . . . . . . . . . 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 6342 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
5142, 50sylan 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑤) ∈ ran 𝑓)
52 sseq2 3619 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑓𝑤) → (𝑧𝑦𝑧 ⊆ (𝑓𝑤)))
5352rspcev 3304 . . . . . . . . . . . . . . . . . . . 20 (((𝑓𝑤) ∈ ran 𝑓𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5451, 53sylan 488 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5554ex 450 . . . . . . . . . . . . . . . . . 18 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑧 ⊆ (𝑓𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5655rexlimdva 3027 . . . . . . . . . . . . . . . . 17 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5756ralimdv 2960 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
5857imp 445 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦)
5958adantl 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)
60 alephislim 8891 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
6160biimpi 206 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → Lim (ℵ‘𝐴))
62 frn 6040 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
6362adantr 481 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
64 coflim 9068 . . . . . . . . . . . . . . 15 ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → ( ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧𝑦))
6561, 63, 64syl2an 494 . . . . . . . . . . . . . 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))
6659, 65mpbird 247 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓𝑤))) → ran 𝑓 = (ℵ‘𝐴))
6749, 66eqtr3d 2656 . . . . . . . . . . . 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))
68 ffvelrn 6343 . . . . . . . . . . . . . . . . 17 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ∈ (ℵ‘𝐴))
6915oneli 5823 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ (ℵ‘𝐴) → (𝑓𝑥) ∈ On)
70 harcard 8789 . . . . . . . . . . . . . . . . . . 19 (card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥))
71 iscard 8786 . . . . . . . . . . . . . . . . . . . 20 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) ↔ ((har‘(𝑓𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))))
7271simprbi 480 . . . . . . . . . . . . . . . . . . 19 ((card‘(har‘(𝑓𝑥))) = (har‘(𝑓𝑥)) → ∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥)))
7370, 72ax-mp 5 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥))
74 domrefg 7975 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ V → (𝑓𝑥) ≼ (𝑓𝑥))
7546, 74ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑥) ≼ (𝑓𝑥)
76 elharval 8453 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) ↔ ((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)))
7776biimpri 218 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑥) ∈ On ∧ (𝑓𝑥) ≼ (𝑓𝑥)) → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
7875, 77mpan2 706 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑥) ∈ On → (𝑓𝑥) ∈ (har‘(𝑓𝑥)))
79 breq1 4647 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑓𝑥) → (𝑦 ≺ (har‘(𝑓𝑥)) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8079rspccv 3301 . . . . . . . . . . . . . . . . . 18 (∀𝑦 ∈ (har‘(𝑓𝑥))𝑦 ≺ (har‘(𝑓𝑥)) → ((𝑓𝑥) ∈ (har‘(𝑓𝑥)) → (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
8173, 78, 80mpsyl 68 . . . . . . . . . . . . . . . . 17 ((𝑓𝑥) ∈ On → (𝑓𝑥) ≺ (har‘(𝑓𝑥)))
8268, 69, 813syl 18 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (har‘(𝑓𝑥)))
83 harcl 8451 . . . . . . . . . . . . . . . . . . 19 (har‘(𝑓𝑥)) ∈ On
84 fveq2 6178 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑥 → (𝑓𝑦) = (𝑓𝑥))
8584fveq2d 6182 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (har‘(𝑓𝑦)) = (har‘(𝑓𝑥)))
86 pwcfsdom.1 . . . . . . . . . . . . . . . . . . . 20 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))
8785, 86fvmptg 6267 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓𝑥)) ∈ On) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8883, 87mpan2 706 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻𝑥) = (har‘(𝑓𝑥)))
8988breq2d 4656 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
9089adantl 482 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓𝑥) ≺ (𝐻𝑥) ↔ (𝑓𝑥) ≺ (har‘(𝑓𝑥))))
9182, 90mpbird 247 . . . . . . . . . . . . . . 15 ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓𝑥) ≺ (𝐻𝑥))
9291ralrimiva 2963 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥))
93 fvex 6188 . . . . . . . . . . . . . . 15 (cf‘(ℵ‘𝐴)) ∈ V
94 eqid 2620 . . . . . . . . . . . . . . 15 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) = 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥)
95 eqid 2620 . . . . . . . . . . . . . . 15 X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥)
9693, 94, 95konigth 9376 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ (𝐻𝑥) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9792, 96syl 17 . . . . . . . . . . . . 13 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥))
9897ad2antrl 763 . . . . . . . . . . . 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))
9967, 98eqbrtrrd 4668 . . . . . . . . . . 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))
10041, 99sylan 488 . . . . . . . . . 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))
101 ovex 6663 . . . . . . . . . . . 12 ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V
10268ex 450 . . . . . . . . . . . . . . . 16 (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓𝑥) ∈ (ℵ‘𝐴)))
103 alephlim 8875 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
104103eleq2d 2685 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦)))
105 eliun 4515 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦))
106 alephcard 8878 . . . . . . . . . . . . . . . . . . . . . . . . 25 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
107106eleq2i 2691 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓𝑥) ∈ (ℵ‘𝑦))
108 cardsdomelir 8784 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓𝑥) ≺ (ℵ‘𝑦))
109107, 108sylbir 225 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (𝑓𝑥) ≺ (ℵ‘𝑦))
110 elharval 8453 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓𝑥)))
111110simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → (ℵ‘𝑦) ≼ (𝑓𝑥))
112 domnsym 8071 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((ℵ‘𝑦) ≼ (𝑓𝑥) → ¬ (𝑓𝑥) ≺ (ℵ‘𝑦))
113111, 112syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℵ‘𝑦) ∈ (har‘(𝑓𝑥)) → ¬ (𝑓𝑥) ≺ (ℵ‘𝑦))
114113con2i 134 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
115 alephon 8877 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ℵ‘𝑦) ∈ On
116 ontri1 5745 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥))))
11783, 115, 116mp2an 707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓𝑥)))
118114, 117sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
119109, 118syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦))
120 alephord2i 8885 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
121120imp 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
122 ontr2 5760 . . . . . . . . . . . . . . . . . . . . . . 23 (((har‘(𝑓𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12383, 15, 122mp2an 707 . . . . . . . . . . . . . . . . . . . . . 22 (((har‘(𝑓𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
124119, 121, 123syl2anr 495 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑦𝐴) ∧ (𝑓𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
125124exp31 629 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ On → (𝑦𝐴 → ((𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))))
126125rexlimdv 3026 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On → (∃𝑦𝐴 (𝑓𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
127105, 126syl5bi 232 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
12841, 127syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ 𝑦𝐴 (ℵ‘𝑦) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
129104, 128sylbid 230 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
130102, 129sylan9r 689 . . . . . . . . . . . . . . 15 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴)))
131130imp 445 . . . . . . . . . . . . . 14 ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓𝑥)) ∈ (ℵ‘𝐴))
13285cbvmptv 4741 . . . . . . . . . . . . . . 15 (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
13386, 132eqtri 2642 . . . . . . . . . . . . . 14 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑥)))
134131, 133fmptd 6371 . . . . . . . . . . . . 13 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
135 ffvelrn 6343 . . . . . . . . . . . . . . 15 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ∈ (ℵ‘𝐴))
136 onelss 5754 . . . . . . . . . . . . . . 15 ((ℵ‘𝐴) ∈ On → ((𝐻𝑥) ∈ (ℵ‘𝐴) → (𝐻𝑥) ⊆ (ℵ‘𝐴)))
13715, 135, 136mpsyl 68 . . . . . . . . . . . . . 14 ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻𝑥) ⊆ (ℵ‘𝐴))
138137ralrimiva 2963 . . . . . . . . . . . . 13 (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴))
139 ss2ixp 7906 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
14093, 10ixpconst 7903 . . . . . . . . . . . . . 14 X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
141139, 140syl6sseq 3643 . . . . . . . . . . . . 13 (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
142134, 138, 1413syl 18 . . . . . . . . . . . 12 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
143 ssdomg 7986 . . . . . . . . . . . 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)))))
144101, 142, 143mpsyl 68 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
145144adantrr 752 . . . . . . . . . 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))))
146 sdomdomtr 8078 . . . . . . . . . 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))))
147100, 145, 146syl2anc 692 . . . . . . . . 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))))
148147expcom 451 . . . . . . . 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)))))
1491483adant2 1078 . . . . . . 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)))))
150149exlimiv 1856 . . . . . 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)))))
15115, 40, 150mp2b 10 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
152151a1i 11 . . . 4 (𝐴 ∈ On → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
15333, 39, 1523jaod 1390 . . 3 (𝐴 ∈ On → ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
1542, 153mpd 15 . 2 (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
155 alephfnon 8873 . . . . 5 ℵ Fn On
156 fndm 5978 . . . . 5 (ℵ Fn On → dom ℵ = On)
157155, 156ax-mp 5 . . . 4 dom ℵ = On
158157eleq2i 2691 . . 3 (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
159 ndmfv 6205 . . . 4 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
160 1n0 7560 . . . . . 6 1𝑜 ≠ ∅
161 1on 7552 . . . . . . . 8 1𝑜 ∈ On
162161elexi 3208 . . . . . . 7 1𝑜 ∈ V
1631620sdom 8076 . . . . . 6 (∅ ≺ 1𝑜 ↔ 1𝑜 ≠ ∅)
164160, 163mpbir 221 . . . . 5 ∅ ≺ 1𝑜
165 id 22 . . . . . 6 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
166 fveq2 6178 . . . . . . . . 9 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
167 cf0 9058 . . . . . . . . 9 (cf‘∅) = ∅
168166, 167syl6eq 2670 . . . . . . . 8 ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
169165, 168oveq12d 6653 . . . . . . 7 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = (∅ ↑𝑚 ∅))
170 0ex 4781 . . . . . . . 8 ∅ ∈ V
171 map0e 7880 . . . . . . . 8 (∅ ∈ V → (∅ ↑𝑚 ∅) = 1𝑜)
172170, 171ax-mp 5 . . . . . . 7 (∅ ↑𝑚 ∅) = 1𝑜
173169, 172syl6eq 2670 . . . . . 6 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = 1𝑜)
174165, 173breq12d 4657 . . . . 5 ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1𝑜))
175164, 174mpbiri 248 . . . 4 ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
176159, 175syl 17 . . 3 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
177158, 176sylnbir 321 . 2 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
178154, 177pm2.61i 176 1 (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wne 2791  wral 2909  wrex 2910  Vcvv 3195  wss 3567  c0 3907  𝒫 cpw 4149   cuni 4427   ciun 4511   class class class wbr 4644  cmpt 4720  dom cdm 5104  ran crn 5105  Oncon0 5711  Lim wlim 5712  suc csuc 5713   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  ωcom 7050  Smo wsmo 7427  1𝑜c1o 7538  2𝑜c2o 7539  𝑚 cmap 7842  Xcixp 7893  cen 7937  cdom 7938  csdm 7939  harchar 8446  cardccrd 8746  cale 8747  cfccf 8748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-smo 7428  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751  df-cf 8752  df-acn 8753  df-ac 8924
This theorem is referenced by:  cfpwsdom  9391  tskcard  9588  bj-pwcfsdom  32997
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