Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axcc3 Structured version   Visualization version   GIF version

Theorem axcc3 9207
 Description: A possibly more useful version of ax-cc 9204 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 7907 . . . 4 Rel ≈
32brrelexi 5120 . . 3 (𝑁 ≈ ω → 𝑁 ∈ V)
4 mptexg 6441 . . 3 (𝑁 ∈ V → (𝑛𝑁𝐹) ∈ V)
51, 3, 4mp2b 10 . 2 (𝑛𝑁𝐹) ∈ V
6 bren 7911 . . . 4 (𝑁 ≈ ω ↔ ∃ :𝑁1-1-onto→ω)
71, 6mpbi 220 . . 3 :𝑁1-1-onto→ω
8 axcc2 9206 . . . . 5 𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
9 f1of 6096 . . . . . . . . . . 11 (:𝑁1-1-onto→ω → :𝑁⟶ω)
10 fnfco 6028 . . . . . . . . . . 11 ((𝑔 Fn ω ∧ :𝑁⟶ω) → (𝑔) Fn 𝑁)
119, 10sylan2 491 . . . . . . . . . 10 ((𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
1211adantlr 750 . . . . . . . . 9 (((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
13123adant1 1077 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑔) Fn 𝑁)
14 nfmpt1 4709 . . . . . . . . . . 11 𝑛(𝑛𝑁𝐹)
1514nfeq2 2776 . . . . . . . . . 10 𝑛 𝑘 = (𝑛𝑁𝐹)
16 nfv 1840 . . . . . . . . . 10 𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))
17 nfv 1840 . . . . . . . . . 10 𝑛 :𝑁1-1-onto→ω
1815, 16, 17nf3an 1828 . . . . . . . . 9 𝑛(𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω)
199ffvelrnda 6317 . . . . . . . . . . . . . . . . . 18 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑛) ∈ ω)
20 fveq2 6150 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → ((𝑘)‘𝑚) = ((𝑘)‘(𝑛)))
2120neeq1d 2849 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → (((𝑘)‘𝑚) ≠ ∅ ↔ ((𝑘)‘(𝑛)) ≠ ∅))
22 fveq2 6150 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = (𝑛) → (𝑔𝑚) = (𝑔‘(𝑛)))
2322, 20eleq12d 2692 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (𝑛) → ((𝑔𝑚) ∈ ((𝑘)‘𝑚) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
2421, 23imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑚 = (𝑛) → ((((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) ↔ (((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)))))
2524rspcv 3291 . . . . . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . . . . 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 6108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:𝑁1-1-onto→ω → :ω–1-1-onto𝑁)
29 f1of 6096 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:ω–1-1-onto𝑁:ω⟶𝑁)
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (:𝑁1-1-onto→ω → :ω⟶𝑁)
31 fvco3 6234 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:ω⟶𝑁 ∧ (𝑛) ∈ ω) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
3230, 31sylan 488 . . . . . . . . . . . . . . . . . . . . . . 23 ((:𝑁1-1-onto→ω ∧ (𝑛) ∈ ω) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
3319, 32syldan 487 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
34333adant1 1077 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = (𝑘‘(‘(𝑛))))
35 f1ocnvfv1 6489 . . . . . . . . . . . . . . . . . . . . . . 23 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (‘(𝑛)) = 𝑛)
3635fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . 22 ((:𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
37363adant1 1077 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘‘(‘(𝑛))) = (𝑘𝑛))
38 fveq1 6149 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑛𝑁𝐹) → (𝑘𝑛) = ((𝑛𝑁𝐹)‘𝑛))
39 axcc3.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐹 ∈ V
40 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛𝑁𝐹) = (𝑛𝑁𝐹)
4140fvmpt2 6250 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝑁𝐹 ∈ V) → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4239, 41mpan2 706 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝑁 → ((𝑛𝑁𝐹)‘𝑛) = 𝐹)
4338, 42sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
44433adant2 1078 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → (𝑘𝑛) = 𝐹)
4534, 37, 443eqtrd 2659 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
46453expa 1262 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
47463adantl2 1216 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑘)‘(𝑛)) = 𝐹)
4847neeq1d 2849 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑘)‘(𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅))
4993ad2ant3 1082 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → :𝑁⟶ω)
50 fvco3 6234 . . . . . . . . . . . . . . . . . . . 20 ((:𝑁⟶ω ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5149, 50sylan 488 . . . . . . . . . . . . . . . . . . 19 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔)‘𝑛) = (𝑔‘(𝑛)))
5251eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))))
5347eleq2d 2684 . . . . . . . . . . . . . . . . . 18 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (((𝑔)‘𝑛) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5452, 53bitr3d 270 . . . . . . . . . . . . . . . . 17 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛)) ↔ ((𝑔)‘𝑛) ∈ 𝐹))
5548, 54imbi12d 334 . . . . . . . . . . . . . . . 16 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → ((((𝑘)‘(𝑛)) ≠ ∅ → (𝑔‘(𝑛)) ∈ ((𝑘)‘(𝑛))) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5627, 55sylibd 229 . . . . . . . . . . . . . . 15 (((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) ∧ 𝑛𝑁) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
5756ex 450 . . . . . . . . . . . . . 14 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (𝑛𝑁 → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
5857com23 86 . . . . . . . . . . . . 13 ((𝑘 = (𝑛𝑁𝐹) ∧ 𝑔 Fn ω ∧ :𝑁1-1-onto→ω) → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
59583exp 1261 . . . . . . . . . . . 12 (𝑘 = (𝑛𝑁𝐹) → (𝑔 Fn ω → (:𝑁1-1-onto→ω → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))))
6059com34 91 . . . . . . . . . . 11 (𝑘 = (𝑛𝑁𝐹) → (𝑔 Fn ω → (∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)) → (:𝑁1-1-onto→ω → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))))
6160imp32 449 . . . . . . . . . 10 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚)))) → (:𝑁1-1-onto→ω → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
62613impia 1258 . . . . . . . . 9 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → (𝑛𝑁 → (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
6318, 62ralrimi 2951 . . . . . . . 8 ((𝑘 = (𝑛𝑁𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘)‘𝑚) ≠ ∅ → (𝑔𝑚) ∈ ((𝑘)‘𝑚))) ∧ :𝑁1-1-onto→ω) → ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))
64 vex 3189 . . . . . . . . . 10 𝑔 ∈ V
65 vex 3189 . . . . . . . . . 10 ∈ V
6664, 65coex 7068 . . . . . . . . 9 (𝑔) ∈ V
67 fneq1 5939 . . . . . . . . . 10 (𝑓 = (𝑔) → (𝑓 Fn 𝑁 ↔ (𝑔) Fn 𝑁))
68 fveq1 6149 . . . . . . . . . . . . 13 (𝑓 = (𝑔) → (𝑓𝑛) = ((𝑔)‘𝑛))
6968eleq1d 2683 . . . . . . . . . . . 12 (𝑓 = (𝑔) → ((𝑓𝑛) ∈ 𝐹 ↔ ((𝑔)‘𝑛) ∈ 𝐹))
7069imbi2d 330 . . . . . . . . . . 11 (𝑓 = (𝑔) → ((𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7170ralbidv 2980 . . . . . . . . . 10 (𝑓 = (𝑔) → (∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹) ↔ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)))
7267, 71anbi12d 746 . . . . . . . . 9 (𝑓 = (𝑔) → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)) ↔ ((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹))))
7366, 72spcev 3286 . . . . . . . 8 (((𝑔) Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → ((𝑔)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
7413, 63, 73syl2anc 692 . . . . . . 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)))
75743exp 1261 . . . . . 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)))))
7675exlimdv 1858 . . . . 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)))))
778, 76mpi 20 . . . 4 (𝑘 = (𝑛𝑁𝐹) → (:𝑁1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))))
7877exlimdv 1858 . . 3 (𝑘 = (𝑛𝑁𝐹) → (∃ :𝑁1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))))
797, 78mpi 20 . 2 (𝑘 = (𝑛𝑁𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹)))
805, 79vtocle 3268 1 𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝐹 ≠ ∅ → (𝑓𝑛) ∈ 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  Vcvv 3186  ∅c0 3893   class class class wbr 4615   ↦ cmpt 4675  ◡ccnv 5075   ∘ ccom 5080   Fn wfn 5844  ⟶wf 5845  –1-1-onto→wf1o 5848  ‘cfv 5849  ωcom 7015   ≈ cen 7899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cc 9204 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-om 7016  df-2nd 7117  df-er 7690  df-en 7903 This theorem is referenced by:  axcc4  9208  domtriomlem  9211  ovnsubaddlem2  40108
 Copyright terms: Public domain W3C validator