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Theorem axcc3 10195
Description: A possibly more useful version of ax-cc 10192 using sequences 𝐹(𝑛) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
axcc3.1 𝐹 ∈ V
axcc3.2 𝑁 ≈ ω
Assertion
Ref Expression
axcc3 𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))
Distinct variable groups:   𝑓,𝐹   𝑓,𝑁,𝑛
Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem axcc3
Dummy variables 𝑔 𝑘 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axcc3.2 . . 3 𝑁 ≈ ω
2 relen 8721 . . . 4 Rel ≈
32brrelex1i 5644 . . 3 (𝑁 ≈ ω → 𝑁 ∈ V)
4 mptexg 7094 . . 3 (𝑁 ∈ V → (𝑛𝑁𝐹) ∈ V)
51, 3, 4mp2b 10 . 2 (𝑛𝑁𝐹) ∈ V
6 bren 8726 . . . 4 (𝑁 ≈ ω ↔ ∃ :𝑁1-1-onto→ω)
71, 6mpbi 229 . . 3 :𝑁1-1-onto→ω
8 axcc2 10194 . . . . 5 𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
9 f1of 6714 . . . . . . . . . . 11 (:𝑁1-1-onto→ω → :𝑁⟶ω)
10 fnfco 6637 . . . . . . . . . . 11 ((𝑔 Fn ω ∧ :𝑁⟶ω) → (𝑔) Fn 𝑁)
119, 10sylan2 593 . . . . . . . . . 10 ((𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
1211adantlr 712 . . . . . . . . 9 (((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
13123adant1 1129 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
14 nfmpt1 5187 . . . . . . . . . . 11 𝑛(𝑛𝑁𝐹)
1514nfeq2 2926 . . . . . . . . . 10 𝑛 𝑘 = (𝑛𝑁𝐹)
16 nfv 1921 . . . . . . . . . 10 𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
17 nfv 1921 . . . . . . . . . 10 𝑛 :𝑁1-1-onto→ω
1815, 16, 17nf3an 1908 . . . . . . . . 9 𝑛(𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω)
199ffvelrnda 6958 . . . . . . . . . . . . . . . . . 18 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑛) ∈ ω)
20 fveq2 6771 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → ((𝑘)‘𝑚) = ((𝑘)‘(𝑛)))
2120neeq1d 3005 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → (((𝑘)‘𝑚) ≠ ∅ ↔ ((𝑘)‘(𝑛)) ≠ ∅))
22 fveq2 6771 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → (𝑔𝑚) = (𝑔‘(𝑛)))
2322, 20eleq12d 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → ((𝑔𝑚) ∈ ((𝑘)‘𝑚) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
2421, 23imbi12d 345 . . . . . . . . . . . . . . . . . . 19 (𝑚 = (𝑛) → ((((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) ↔ (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
2524rspcv 3556 . . . . . . . . . . . . . . . . . 18 ((𝑛) ∈ ω → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
2619, 25syl 17 . . . . . . . . . . . . . . . . 17 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
27263ad2antl3 1186 . . . . . . . . . . . . . . . 16 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
28 f1ocnv 6726 . . . . . . . . . . . . . . . . . . . . . . . 24 (:𝑁1-1-onto→ω → :ω–1-1-onto𝑁)
29 f1of 6714 . . . . . . . . . . . . . . . . . . . . . . . 24 (:ω–1-1-onto𝑁:ω⟶𝑁)
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (:𝑁1-1-onto→ω → :ω⟶𝑁)
31 fvco3 6864 . . . . . . . . . . . . . . . . . . . . . . 23 ((:ω⟶𝑁 ∧ (𝑛) ∈ ω) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
3230, 19, 31syl2an2r 682 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
33323adant1 1129 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
34 f1ocnvfv1 7145 . . . . . . . . . . . . . . . . . . . . . . 23 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (‘(𝑛)) = 𝑛)
3534fveq2d 6775 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
36353adant1 1129 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
37 fveq1 6770 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑛𝑁𝐹) → (𝑘𝑛) = ((𝑛𝑁𝐹)‘𝑛))
38 axcc3.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐹 ∈ V
39 eqid 2740 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛𝑁𝐹) = (𝑛𝑁𝐹)
4039fvmpt2 6883 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝑁𝐹 ∈ V) → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4138, 40mpan2 688 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝑁 → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4237, 41sylan9eq 2800 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
43423adant2 1130 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
4433, 36, 433eqtrd 2784 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
45443expa 1117 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
46453adantl2 1166 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
4746neeq1d 3005 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑘)‘(𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅))
4893ad2ant3 1134 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → :𝑁⟶ω)
49 fvco3 6864 . . . . . . . . . . . . . . . . . . . 20 ((:𝑁⟶ω ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5048, 49sylan 580 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5150eleq1d 2825 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
5246eleq2d 2826 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5351, 52bitr3d 280 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5447, 53imbi12d 345 . . . . . . . . . . . . . . . 16 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5527, 54sylibd 238 . . . . . . . . . . . . . . 15 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5655ex 413 . . . . . . . . . . . . . 14 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (𝑛𝑁 → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
5756com23 86 . . . . . . . . . . . . 13 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
58573exp 1118 . . . . . . . . . . . 12 (𝑘 = (𝑛𝑁𝐹) → (𝑔 Fn ω → (:𝑁1-1-onto→ω → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))))
5958com34 91 . . . . . . . . . . 11 (𝑘 = (𝑛𝑁𝐹) → (𝑔 Fn ω → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (:𝑁1-1-onto→ω → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))))
6059imp32 419 . . . . . . . . . 10 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))) → (:𝑁1-1-onto→ω → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
61603impia 1116 . . . . . . . . 9 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
6218, 61ralrimi 3142 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))
63 vex 3435 . . . . . . . . . 10 𝑔 ∈ V
64 vex 3435 . . . . . . . . . 10 ∈ V
6563, 64coex 7771 . . . . . . . . 9 (𝑔) ∈ V
66 fneq1 6522 . . . . . . . . . 10 (𝑓 = (𝑔) → (𝑓 Fn 𝑁 ↔ (𝑔) Fn 𝑁))
67 fveq1 6770 . . . . . . . . . . . . 13 (𝑓 = (𝑔) → (𝑓𝑛) = ((𝑔)‘𝑛))
6867eleq1d 2825 . . . . . . . . . . . 12 (𝑓 = (𝑔) → ((𝑓𝑛) ∈ 𝐹 ↔ ((𝑔)‘𝑛) ∈ 𝐹))
6968imbi2d 341 . . . . . . . . . . 11 (𝑓 = (𝑔) → ((𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7069ralbidv 3123 . . . . . . . . . 10 (𝑓 = (𝑔) → (∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7166, 70anbi12d 631 . . . . . . . . 9 (𝑓 = (𝑔) → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)) ↔ ((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
7265, 71spcev 3544 . . . . . . . 8 (((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
7313, 62, 72syl2anc 584 . . . . . . 7 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
74733exp 1118 . . . . . 6 (𝑘 = (𝑛𝑁𝐹) → ((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) → (:𝑁1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))))
7574exlimdv 1940 . . . . 5 (𝑘 = (𝑛𝑁𝐹) → (∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) → (:𝑁1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))))
768, 75mpi 20 . . . 4 (𝑘 = (𝑛𝑁𝐹) → (:𝑁1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))))
7776exlimdv 1940 . . 3 (𝑘 = (𝑛𝑁𝐹) → (∃ :𝑁1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))))
787, 77mpi 20 . 2 (𝑘 = (𝑛𝑁𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
795, 78vtocle 3523 1 𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1542  wex 1786  wcel 2110  wne 2945  wral 3066  Vcvv 3431  c0 4262   class class class wbr 5079  cmpt 5162  ccnv 5589  ccom 5594   Fn wfn 6427  wf 6428  1-1-ontowf1o 6431  cfv 6432  ωcom 7706  cen 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-cc 10192
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-om 7707  df-2nd 7825  df-er 8481  df-en 8717
This theorem is referenced by:  axcc4  10196  domtriomlem  10199  ovnsubaddlem2  44080
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