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Theorem axcc3 9658
Description: A possibly more useful version of ax-cc 9655 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 8311 . . . 4 Rel ≈
32brrelex1i 5458 . . 3 (𝑁 ≈ ω → 𝑁 ∈ V)
4 mptexg 6810 . . 3 (𝑁 ∈ V → (𝑛𝑁𝐹) ∈ V)
51, 3, 4mp2b 10 . 2 (𝑛𝑁𝐹) ∈ V
6 bren 8315 . . . 4 (𝑁 ≈ ω ↔ ∃ :𝑁1-1-onto→ω)
71, 6mpbi 222 . . 3 :𝑁1-1-onto→ω
8 axcc2 9657 . . . . 5 𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
9 f1of 6444 . . . . . . . . . . 11 (:𝑁1-1-onto→ω → :𝑁⟶ω)
10 fnfco 6372 . . . . . . . . . . 11 ((𝑔 Fn ω ∧ :𝑁⟶ω) → (𝑔) Fn 𝑁)
119, 10sylan2 583 . . . . . . . . . 10 ((𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
1211adantlr 702 . . . . . . . . 9 (((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
13123adant1 1110 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
14 nfmpt1 5025 . . . . . . . . . . 11 𝑛(𝑛𝑁𝐹)
1514nfeq2 2948 . . . . . . . . . 10 𝑛 𝑘 = (𝑛𝑁𝐹)
16 nfv 1873 . . . . . . . . . 10 𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
17 nfv 1873 . . . . . . . . . 10 𝑛 :𝑁1-1-onto→ω
1815, 16, 17nf3an 1864 . . . . . . . . 9 𝑛(𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω)
199ffvelrnda 6676 . . . . . . . . . . . . . . . . . 18 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑛) ∈ ω)
20 fveq2 6499 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → ((𝑘)‘𝑚) = ((𝑘)‘(𝑛)))
2120neeq1d 3027 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → (((𝑘)‘𝑚) ≠ ∅ ↔ ((𝑘)‘(𝑛)) ≠ ∅))
22 fveq2 6499 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → (𝑔𝑚) = (𝑔‘(𝑛)))
2322, 20eleq12d 2861 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → ((𝑔𝑚) ∈ ((𝑘)‘𝑚) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
2421, 23imbi12d 337 . . . . . . . . . . . . . . . . . . 19 (𝑚 = (𝑛) → ((((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) ↔ (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
2524rspcv 3532 . . . . . . . . . . . . . . . . . 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 1167 . . . . . . . . . . . . . . . 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 6456 . . . . . . . . . . . . . . . . . . . . . . . 24 (:𝑁1-1-onto→ω → :ω–1-1-onto𝑁)
29 f1of 6444 . . . . . . . . . . . . . . . . . . . . . . . 24 (:ω–1-1-onto𝑁:ω⟶𝑁)
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (:𝑁1-1-onto→ω → :ω⟶𝑁)
31 fvco3 6588 . . . . . . . . . . . . . . . . . . . . . . 23 ((:ω⟶𝑁 ∧ (𝑛) ∈ ω) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
3230, 19, 31syl2an2r 672 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
33323adant1 1110 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
34 f1ocnvfv1 6858 . . . . . . . . . . . . . . . . . . . . . . 23 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (‘(𝑛)) = 𝑛)
3534fveq2d 6503 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
36353adant1 1110 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
37 fveq1 6498 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑛𝑁𝐹) → (𝑘𝑛) = ((𝑛𝑁𝐹)‘𝑛))
38 axcc3.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐹 ∈ V
39 eqid 2779 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛𝑁𝐹) = (𝑛𝑁𝐹)
4039fvmpt2 6605 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝑁𝐹 ∈ V) → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4138, 40mpan2 678 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝑁 → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4237, 41sylan9eq 2835 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
43423adant2 1111 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
4433, 36, 433eqtrd 2819 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
45443expa 1098 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
46453adantl2 1147 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
4746neeq1d 3027 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑘)‘(𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅))
4893ad2ant3 1115 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → :𝑁⟶ω)
49 fvco3 6588 . . . . . . . . . . . . . . . . . . . 20 ((:𝑁⟶ω ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5048, 49sylan 572 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5150eleq1d 2851 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
5246eleq2d 2852 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5351, 52bitr3d 273 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5447, 53imbi12d 337 . . . . . . . . . . . . . . . 16 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5527, 54sylibd 231 . . . . . . . . . . . . . . 15 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5655ex 405 . . . . . . . . . . . . . 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 1099 . . . . . . . . . . . 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 411 . . . . . . . . . 10 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))) → (:𝑁1-1-onto→ω → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
61603impia 1097 . . . . . . . . 9 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
6218, 61ralrimi 3167 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))
63 vex 3419 . . . . . . . . . 10 𝑔 ∈ V
64 vex 3419 . . . . . . . . . 10 ∈ V
6563, 64coex 7450 . . . . . . . . 9 (𝑔) ∈ V
66 fneq1 6277 . . . . . . . . . 10 (𝑓 = (𝑔) → (𝑓 Fn 𝑁 ↔ (𝑔) Fn 𝑁))
67 fveq1 6498 . . . . . . . . . . . . 13 (𝑓 = (𝑔) → (𝑓𝑛) = ((𝑔)‘𝑛))
6867eleq1d 2851 . . . . . . . . . . . 12 (𝑓 = (𝑔) → ((𝑓𝑛) ∈ 𝐹 ↔ ((𝑔)‘𝑛) ∈ 𝐹))
6968imbi2d 333 . . . . . . . . . . 11 (𝑓 = (𝑔) → ((𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7069ralbidv 3148 . . . . . . . . . 10 (𝑓 = (𝑔) → (∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7166, 70anbi12d 621 . . . . . . . . 9 (𝑓 = (𝑔) → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)) ↔ ((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
7265, 71spcev 3526 . . . . . . . 8 (((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
7313, 62, 72syl2anc 576 . . . . . . 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 1099 . . . . . 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 1892 . . . . 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 1892 . . 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 3504 1 𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wex 1742  wcel 2050  wne 2968  wral 3089  Vcvv 3416  c0 4179   class class class wbr 4929  cmpt 5008  ccnv 5406  ccom 5411   Fn wfn 6183  wf 6184  1-1-ontowf1o 6187  cfv 6188  ωcom 7396  cen 8303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898  ax-cc 9655
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-2nd 7502  df-er 8089  df-en 8307
This theorem is referenced by:  axcc4  9659  domtriomlem  9662  ovnsubaddlem2  42282
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