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Theorem pwfseq 10665
Description: The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
pwfseq (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
Distinct variable group:   𝐴,𝑛

Proof of Theorem pwfseq
Dummy variables 𝑓 𝑏 𝑔 𝑘 𝑚 𝑝 𝑟 𝑠 𝑡 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 8951 . . 3 Rel ≼
21brrelex2i 5733 . 2 (ω ≼ 𝐴𝐴 ∈ V)
3 domeng 8964 . . 3 (𝐴 ∈ V → (ω ≼ 𝐴 ↔ ∃𝑡(ω ≈ 𝑡𝑡𝐴)))
4 bren 8955 . . . . . 6 (ω ≈ 𝑡 ↔ ∃ :ω–1-1-onto𝑡)
5 harcl 9560 . . . . . . . . . 10 (har‘𝒫 𝐴) ∈ On
6 infxpenc2 10023 . . . . . . . . . 10 ((har‘𝒫 𝐴) ∈ On → ∃𝑚𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
75, 6ax-mp 5 . . . . . . . . 9 𝑚𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)
8 simpr 484 . . . . . . . . . . . . . . . 16 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛)) → 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
9 oveq2 7420 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → (𝐴m 𝑛) = (𝐴m 𝑘))
109cbviunv 5043 . . . . . . . . . . . . . . . . 17 𝑛 ∈ ω (𝐴m 𝑛) = 𝑘 ∈ ω (𝐴m 𝑘)
11 f1eq3 6784 . . . . . . . . . . . . . . . . 17 ( 𝑛 ∈ ω (𝐴m 𝑛) = 𝑘 ∈ ω (𝐴m 𝑘) → (𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛) ↔ 𝑔:𝒫 𝐴1-1 𝑘 ∈ ω (𝐴m 𝑘)))
1210, 11ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛) ↔ 𝑔:𝒫 𝐴1-1 𝑘 ∈ ω (𝐴m 𝑘))
138, 12sylib 217 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛)) → 𝑔:𝒫 𝐴1-1 𝑘 ∈ ω (𝐴m 𝑘))
14 simpllr 773 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛)) → 𝑡𝐴)
15 simplll 772 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛)) → :ω–1-1-onto𝑡)
16 biid 261 . . . . . . . . . . . . . . 15 (((𝑢𝐴𝑟 ⊆ (𝑢 × 𝑢) ∧ 𝑟 We 𝑢) ∧ ω ≼ 𝑢) ↔ ((𝑢𝐴𝑟 ⊆ (𝑢 × 𝑢) ∧ 𝑟 We 𝑢) ∧ ω ≼ 𝑢))
17 simplr 766 . . . . . . . . . . . . . . . 16 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛)) → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
18 sseq2 4008 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑤 → (ω ⊆ 𝑏 ↔ ω ⊆ 𝑤))
19 fveq2 6891 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑤 → (𝑚𝑏) = (𝑚𝑤))
2019f1oeq1d 6828 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑤 → ((𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑏 × 𝑏)–1-1-onto𝑏))
21 xpeq12 5701 . . . . . . . . . . . . . . . . . . . . 21 ((𝑏 = 𝑤𝑏 = 𝑤) → (𝑏 × 𝑏) = (𝑤 × 𝑤))
2221anidms 566 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑤 → (𝑏 × 𝑏) = (𝑤 × 𝑤))
2322f1oeq2d 6829 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑤 → ((𝑚𝑤):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑏))
24 f1oeq3 6823 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑤 → ((𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
2520, 23, 243bitrd 305 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑤 → ((𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
2618, 25imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑤 → ((ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ (ω ⊆ 𝑤 → (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤)))
2726cbvralvw 3233 . . . . . . . . . . . . . . . 16 (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ ∀𝑤 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑤 → (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
2817, 27sylib 217 . . . . . . . . . . . . . . 15 ((((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛)) → ∀𝑤 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑤 → (𝑚𝑤):(𝑤 × 𝑤)–1-1-onto𝑤))
29 eqid 2731 . . . . . . . . . . . . . . 15 OrdIso(𝑟, 𝑢) = OrdIso(𝑟, 𝑢)
30 eqid 2731 . . . . . . . . . . . . . . 15 (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩) = (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)
31 eqid 2731 . . . . . . . . . . . . . . 15 ((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)) = ((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))
32 eqid 2731 . . . . . . . . . . . . . . 15 seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢m suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩}) = seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢m suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})
33 oveq2 7420 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (𝑢m 𝑛) = (𝑢m 𝑘))
3433cbviunv 5043 . . . . . . . . . . . . . . . 16 𝑛 ∈ ω (𝑢m 𝑛) = 𝑘 ∈ ω (𝑢m 𝑘)
3534mpteq1i 5244 . . . . . . . . . . . . . . 15 (𝑦 𝑛 ∈ ω (𝑢m 𝑛) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢m suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩) = (𝑦 𝑘 ∈ ω (𝑢m 𝑘) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢m suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩)
36 eqid 2731 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩) = (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)
37 eqid 2731 . . . . . . . . . . . . . . 15 ((((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)) ∘ (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)) ∘ (𝑦 𝑛 ∈ ω (𝑢m 𝑛) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢m suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩)) = ((((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩)) ∘ (𝑥 ∈ ω, 𝑦𝑢 ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑥), 𝑦⟩)) ∘ (𝑦 𝑛 ∈ ω (𝑢m 𝑛) ↦ ⟨dom 𝑦, ((seqω((𝑝 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑢m suc 𝑝) ↦ ((𝑓‘(𝑥𝑝))((OrdIso(𝑟, 𝑢) ∘ (𝑚‘dom OrdIso(𝑟, 𝑢))) ∘ (𝑠 ∈ dom OrdIso(𝑟, 𝑢), 𝑧 ∈ dom OrdIso(𝑟, 𝑢) ↦ ⟨(OrdIso(𝑟, 𝑢)‘𝑠), (OrdIso(𝑟, 𝑢)‘𝑧)⟩))(𝑥𝑝)))), {⟨∅, (OrdIso(𝑟, 𝑢)‘∅)⟩})‘dom 𝑦)‘𝑦)⟩))
3813, 14, 15, 16, 28, 29, 30, 31, 32, 35, 36, 37pwfseqlem5 10664 . . . . . . . . . . . . . 14 ¬ (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) ∧ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
3938imnani 400 . . . . . . . . . . . . 13 (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) → ¬ 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
4039nexdv 1938 . . . . . . . . . . . 12 (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) → ¬ ∃𝑔 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
41 brdomi 8960 . . . . . . . . . . . 12 (𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛) → ∃𝑔 𝑔:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))
4240, 41nsyl 140 . . . . . . . . . . 11 (((:ω–1-1-onto𝑡𝑡𝐴) ∧ ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
4342ex 412 . . . . . . . . . 10 ((:ω–1-1-onto𝑡𝑡𝐴) → (∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
4443exlimdv 1935 . . . . . . . . 9 ((:ω–1-1-onto𝑡𝑡𝐴) → (∃𝑚𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑚𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
457, 44mpi 20 . . . . . . . 8 ((:ω–1-1-onto𝑡𝑡𝐴) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
4645ex 412 . . . . . . 7 (:ω–1-1-onto𝑡 → (𝑡𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
4746exlimiv 1932 . . . . . 6 (∃ :ω–1-1-onto𝑡 → (𝑡𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
484, 47sylbi 216 . . . . 5 (ω ≈ 𝑡 → (𝑡𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
4948imp 406 . . . 4 ((ω ≈ 𝑡𝑡𝐴) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
5049exlimiv 1932 . . 3 (∃𝑡(ω ≈ 𝑡𝑡𝐴) → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
513, 50syl6bi 253 . 2 (𝐴 ∈ V → (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛)))
522, 51mpcom 38 1 (ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴m 𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wral 3060  Vcvv 3473  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628  cop 4634   ciun 4997   class class class wbr 5148  cmpt 5231   We wwe 5630   × cxp 5674  ccnv 5675  dom cdm 5676  cres 5678  ccom 5680  Oncon0 6364  suc csuc 6366  1-1wf1 6540  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7412  cmpo 7414  ωcom 7859  seqωcseqom 8453  m cmap 8826  cen 8942  cdom 8943  OrdIsocoi 9510  harchar 9557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-seqom 8454  df-1o 8472  df-2o 8473  df-oadd 8476  df-omul 8477  df-oexp 8478  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-oi 9511  df-har 9558  df-cnf 9663  df-card 9940
This theorem is referenced by:  pwxpndom2  10666
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