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Theorem axcc3 9853
Description: A possibly more useful version of ax-cc 9850 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 8501 . . . 4 Rel ≈
32brrelex1i 5576 . . 3 (𝑁 ≈ ω → 𝑁 ∈ V)
4 mptexg 6965 . . 3 (𝑁 ∈ V → (𝑛𝑁𝐹) ∈ V)
51, 3, 4mp2b 10 . 2 (𝑛𝑁𝐹) ∈ V
6 bren 8505 . . . 4 (𝑁 ≈ ω ↔ ∃ :𝑁1-1-onto→ω)
71, 6mpbi 233 . . 3 :𝑁1-1-onto→ω
8 axcc2 9852 . . . . 5 𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
9 f1of 6594 . . . . . . . . . . 11 (:𝑁1-1-onto→ω → :𝑁⟶ω)
10 fnfco 6521 . . . . . . . . . . 11 ((𝑔 Fn ω ∧ :𝑁⟶ω) → (𝑔) Fn 𝑁)
119, 10sylan2 595 . . . . . . . . . 10 ((𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
1211adantlr 714 . . . . . . . . 9 (((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
13123adant1 1127 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
14 nfmpt1 5131 . . . . . . . . . . 11 𝑛(𝑛𝑁𝐹)
1514nfeq2 2975 . . . . . . . . . 10 𝑛 𝑘 = (𝑛𝑁𝐹)
16 nfv 1915 . . . . . . . . . 10 𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
17 nfv 1915 . . . . . . . . . 10 𝑛 :𝑁1-1-onto→ω
1815, 16, 17nf3an 1902 . . . . . . . . 9 𝑛(𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω)
199ffvelrnda 6832 . . . . . . . . . . . . . . . . . 18 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑛) ∈ ω)
20 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → ((𝑘)‘𝑚) = ((𝑘)‘(𝑛)))
2120neeq1d 3049 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → (((𝑘)‘𝑚) ≠ ∅ ↔ ((𝑘)‘(𝑛)) ≠ ∅))
22 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → (𝑔𝑚) = (𝑔‘(𝑛)))
2322, 20eleq12d 2887 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → ((𝑔𝑚) ∈ ((𝑘)‘𝑚) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
2421, 23imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑚 = (𝑛) → ((((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) ↔ (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
2524rspcv 3569 . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 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 6606 . . . . . . . . . . . . . . . . . . . . . . . 24 (:𝑁1-1-onto→ω → :ω–1-1-onto𝑁)
29 f1of 6594 . . . . . . . . . . . . . . . . . . . . . . . 24 (:ω–1-1-onto𝑁:ω⟶𝑁)
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (:𝑁1-1-onto→ω → :ω⟶𝑁)
31 fvco3 6741 . . . . . . . . . . . . . . . . . . . . . . 23 ((:ω⟶𝑁 ∧ (𝑛) ∈ ω) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
3230, 19, 31syl2an2r 684 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
33323adant1 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
34 f1ocnvfv1 7015 . . . . . . . . . . . . . . . . . . . . . . 23 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (‘(𝑛)) = 𝑛)
3534fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
36353adant1 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
37 fveq1 6648 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑛𝑁𝐹) → (𝑘𝑛) = ((𝑛𝑁𝐹)‘𝑛))
38 axcc3.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐹 ∈ V
39 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛𝑁𝐹) = (𝑛𝑁𝐹)
4039fvmpt2 6760 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝑁𝐹 ∈ V) → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4138, 40mpan2 690 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝑁 → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4237, 41sylan9eq 2856 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
43423adant2 1128 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
4433, 36, 433eqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
45443expa 1115 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
46453adantl2 1164 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
4746neeq1d 3049 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑘)‘(𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅))
4893ad2ant3 1132 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → :𝑁⟶ω)
49 fvco3 6741 . . . . . . . . . . . . . . . . . . . 20 ((:𝑁⟶ω ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5048, 49sylan 583 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5150eleq1d 2877 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
5246eleq2d 2878 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5351, 52bitr3d 284 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5447, 53imbi12d 348 . . . . . . . . . . . . . . . 16 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5527, 54sylibd 242 . . . . . . . . . . . . . . 15 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5655ex 416 . . . . . . . . . . . . . 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 1116 . . . . . . . . . . . 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 422 . . . . . . . . . 10 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))) → (:𝑁1-1-onto→ω → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
61603impia 1114 . . . . . . . . 9 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
6218, 61ralrimi 3183 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))
63 vex 3447 . . . . . . . . . 10 𝑔 ∈ V
64 vex 3447 . . . . . . . . . 10 ∈ V
6563, 64coex 7621 . . . . . . . . 9 (𝑔) ∈ V
66 fneq1 6418 . . . . . . . . . 10 (𝑓 = (𝑔) → (𝑓 Fn 𝑁 ↔ (𝑔) Fn 𝑁))
67 fveq1 6648 . . . . . . . . . . . . 13 (𝑓 = (𝑔) → (𝑓𝑛) = ((𝑔)‘𝑛))
6867eleq1d 2877 . . . . . . . . . . . 12 (𝑓 = (𝑔) → ((𝑓𝑛) ∈ 𝐹 ↔ ((𝑔)‘𝑛) ∈ 𝐹))
6968imbi2d 344 . . . . . . . . . . 11 (𝑓 = (𝑔) → ((𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7069ralbidv 3165 . . . . . . . . . 10 (𝑓 = (𝑔) → (∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7166, 70anbi12d 633 . . . . . . . . 9 (𝑓 = (𝑔) → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)) ↔ ((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
7265, 71spcev 3558 . . . . . . . 8 (((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
7313, 62, 72syl2anc 587 . . . . . . 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 1116 . . . . . 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 1934 . . . . 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 1934 . . 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 3535 1 𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  wne 2990  wral 3109  Vcvv 3444  c0 4246   class class class wbr 5033  cmpt 5113  ccnv 5522  ccom 5527   Fn wfn 6323  wf 6324  1-1-ontowf1o 6327  cfv 6328  ωcom 7564  cen 8493
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-cc 9850
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-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-iun 4886  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-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-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-om 7565  df-2nd 7676  df-er 8276  df-en 8497
This theorem is referenced by:  axcc4  9854  domtriomlem  9857  ovnsubaddlem2  43197
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