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Theorem fin23lem28 9106
Description: Lemma for fin23 9155. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem28 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem28
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fin23lem.e . . 3 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4098 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
31, 2mpbi 220 . 2 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)))
4 difss 3715 . . . . . . . . 9 (ω ∖ 𝑃) ⊆ ω
5 ominf 8116 . . . . . . . . . 10 ¬ ω ∈ Fin
6 fin23lem.b . . . . . . . . . . . . . 14 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
7 ssrab2 3666 . . . . . . . . . . . . . 14 {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ⊆ ω
86, 7eqsstri 3614 . . . . . . . . . . . . 13 𝑃 ⊆ ω
9 undif 4021 . . . . . . . . . . . . 13 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
108, 9mpbi 220 . . . . . . . . . . . 12 (𝑃 ∪ (ω ∖ 𝑃)) = ω
11 unfi 8171 . . . . . . . . . . . 12 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
1210, 11syl5eqelr 2703 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
1312ex 450 . . . . . . . . . 10 (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
145, 13mtoi 190 . . . . . . . . 9 (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
15 fin23lem.d . . . . . . . . . 10 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
1615fin23lem22 9093 . . . . . . . . 9 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
174, 14, 16sylancr 694 . . . . . . . 8 (𝑃 ∈ Fin → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
1817adantl 482 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
19 f1of1 6093 . . . . . . 7 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω–1-1→(ω ∖ 𝑃))
20 f1ss 6063 . . . . . . . 8 ((𝑅:ω–1-1→(ω ∖ 𝑃) ∧ (ω ∖ 𝑃) ⊆ ω) → 𝑅:ω–1-1→ω)
214, 20mpan2 706 . . . . . . 7 (𝑅:ω–1-1→(ω ∖ 𝑃) → 𝑅:ω–1-1→ω)
2218, 19, 213syl 18 . . . . . 6 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1→ω)
23 f1co 6067 . . . . . 6 ((𝑡:ω–1-1→V ∧ 𝑅:ω–1-1→ω) → (𝑡𝑅):ω–1-1→V)
2422, 23syldan 487 . . . . 5 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑡𝑅):ω–1-1→V)
25 f1eq1 6053 . . . . 5 (𝑍 = (𝑡𝑅) → (𝑍:ω–1-1→V ↔ (𝑡𝑅):ω–1-1→V))
2624, 25syl5ibrcom 237 . . . 4 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑍 = (𝑡𝑅) → 𝑍:ω–1-1→V))
2726impr 648 . . 3 ((𝑡:ω–1-1→V ∧ (𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅))) → 𝑍:ω–1-1→V)
28 fvex 6158 . . . . . . . . . . 11 (𝑡𝑧) ∈ V
29 difexg 4768 . . . . . . . . . . 11 ((𝑡𝑧) ∈ V → ((𝑡𝑧) ∖ ran 𝑈) ∈ V)
3028, 29ax-mp 5 . . . . . . . . . 10 ((𝑡𝑧) ∖ ran 𝑈) ∈ V
3130rgenw 2919 . . . . . . . . 9 𝑧𝑃 ((𝑡𝑧) ∖ ran 𝑈) ∈ V
32 eqid 2621 . . . . . . . . . 10 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
3332fmpt 6337 . . . . . . . . 9 (∀𝑧𝑃 ((𝑡𝑧) ∖ ran 𝑈) ∈ V ↔ (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V)
3431, 33mpbi 220 . . . . . . . 8 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V
3534a1i 11 . . . . . . 7 (𝑡:ω–1-1→V → (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V)
36 fveq2 6148 . . . . . . . . . . . . 13 (𝑧 = 𝑎 → (𝑡𝑧) = (𝑡𝑎))
3736difeq1d 3705 . . . . . . . . . . . 12 (𝑧 = 𝑎 → ((𝑡𝑧) ∖ ran 𝑈) = ((𝑡𝑎) ∖ ran 𝑈))
38 fvex 6158 . . . . . . . . . . . . 13 (𝑡𝑎) ∈ V
39 difexg 4768 . . . . . . . . . . . . 13 ((𝑡𝑎) ∈ V → ((𝑡𝑎) ∖ ran 𝑈) ∈ V)
4038, 39ax-mp 5 . . . . . . . . . . . 12 ((𝑡𝑎) ∖ ran 𝑈) ∈ V
4137, 32, 40fvmpt 6239 . . . . . . . . . . 11 (𝑎𝑃 → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑡𝑎) ∖ ran 𝑈))
4241ad2antrl 763 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑡𝑎) ∖ ran 𝑈))
43 fveq2 6148 . . . . . . . . . . . . 13 (𝑧 = 𝑏 → (𝑡𝑧) = (𝑡𝑏))
4443difeq1d 3705 . . . . . . . . . . . 12 (𝑧 = 𝑏 → ((𝑡𝑧) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈))
45 fvex 6158 . . . . . . . . . . . . 13 (𝑡𝑏) ∈ V
46 difexg 4768 . . . . . . . . . . . . 13 ((𝑡𝑏) ∈ V → ((𝑡𝑏) ∖ ran 𝑈) ∈ V)
4745, 46ax-mp 5 . . . . . . . . . . . 12 ((𝑡𝑏) ∖ ran 𝑈) ∈ V
4844, 32, 47fvmpt 6239 . . . . . . . . . . 11 (𝑏𝑃 → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) = ((𝑡𝑏) ∖ ran 𝑈))
4948ad2antll 764 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) = ((𝑡𝑏) ∖ ran 𝑈))
5042, 49eqeq12d 2636 . . . . . . . . 9 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) ↔ ((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈)))
51 uneq2 3739 . . . . . . . . . . 11 (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)))
52 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑎 → (𝑡𝑣) = (𝑡𝑎))
5352sseq2d 3612 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑎 → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡𝑎)))
5453, 6elrab2 3348 . . . . . . . . . . . . . . 15 (𝑎𝑃 ↔ (𝑎 ∈ ω ∧ ran 𝑈 ⊆ (𝑡𝑎)))
5554simprbi 480 . . . . . . . . . . . . . 14 (𝑎𝑃 ran 𝑈 ⊆ (𝑡𝑎))
5655ad2antrl 763 . . . . . . . . . . . . 13 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ran 𝑈 ⊆ (𝑡𝑎))
57 undif 4021 . . . . . . . . . . . . 13 ( ran 𝑈 ⊆ (𝑡𝑎) ↔ ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = (𝑡𝑎))
5856, 57sylib 208 . . . . . . . . . . . 12 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = (𝑡𝑎))
59 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑏 → (𝑡𝑣) = (𝑡𝑏))
6059sseq2d 3612 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑏 → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡𝑏)))
6160, 6elrab2 3348 . . . . . . . . . . . . . . 15 (𝑏𝑃 ↔ (𝑏 ∈ ω ∧ ran 𝑈 ⊆ (𝑡𝑏)))
6261simprbi 480 . . . . . . . . . . . . . 14 (𝑏𝑃 ran 𝑈 ⊆ (𝑡𝑏))
6362ad2antll 764 . . . . . . . . . . . . 13 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ran 𝑈 ⊆ (𝑡𝑏))
64 undif 4021 . . . . . . . . . . . . 13 ( ran 𝑈 ⊆ (𝑡𝑏) ↔ ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)) = (𝑡𝑏))
6563, 64sylib 208 . . . . . . . . . . . 12 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)) = (𝑡𝑏))
6658, 65eqeq12d 2636 . . . . . . . . . . 11 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)) ↔ (𝑡𝑎) = (𝑡𝑏)))
6751, 66syl5ib 234 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → (𝑡𝑎) = (𝑡𝑏)))
688sseli 3579 . . . . . . . . . . . 12 (𝑎𝑃𝑎 ∈ ω)
698sseli 3579 . . . . . . . . . . . 12 (𝑏𝑃𝑏 ∈ ω)
7068, 69anim12i 589 . . . . . . . . . . 11 ((𝑎𝑃𝑏𝑃) → (𝑎 ∈ ω ∧ 𝑏 ∈ ω))
71 f1fveq 6473 . . . . . . . . . . 11 ((𝑡:ω–1-1→V ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝑡𝑎) = (𝑡𝑏) ↔ 𝑎 = 𝑏))
7270, 71sylan2 491 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑡𝑎) = (𝑡𝑏) ↔ 𝑎 = 𝑏))
7367, 72sylibd 229 . . . . . . . . 9 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → 𝑎 = 𝑏))
7450, 73sylbid 230 . . . . . . . 8 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
7574ralrimivva 2965 . . . . . . 7 (𝑡:ω–1-1→V → ∀𝑎𝑃𝑏𝑃 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
76 dff13 6466 . . . . . . 7 ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V ↔ ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V ∧ ∀𝑎𝑃𝑏𝑃 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏)))
7735, 75, 76sylanbrc 697 . . . . . 6 (𝑡:ω–1-1→V → (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V)
78 fin23lem.c . . . . . . . . 9 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
7978fin23lem22 9093 . . . . . . . 8 ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1-onto𝑃)
80 f1of1 6093 . . . . . . . 8 (𝑄:ω–1-1-onto𝑃𝑄:ω–1-1𝑃)
8179, 80syl 17 . . . . . . 7 ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1𝑃)
828, 81mpan 705 . . . . . 6 𝑃 ∈ Fin → 𝑄:ω–1-1𝑃)
83 f1co 6067 . . . . . 6 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V ∧ 𝑄:ω–1-1𝑃) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
8477, 82, 83syl2an 494 . . . . 5 ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
85 f1eq1 6053 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → (𝑍:ω–1-1→V ↔ ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V))
8684, 85syl5ibrcom 237 . . . 4 ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → 𝑍:ω–1-1→V))
8786impr 648 . . 3 ((𝑡:ω–1-1→V ∧ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → 𝑍:ω–1-1→V)
8827, 87jaodan 825 . 2 ((𝑡:ω–1-1→V ∧ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)))) → 𝑍:ω–1-1→V)
893, 88mpan2 706 1 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  {crab 2911  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130   cuni 4402   cint 4440   class class class wbr 4613  cmpt 4673  ran crn 5075  ccom 5078  suc csuc 5684  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  crio 6564  (class class class)co 6604  cmpt2 6606  ωcom 7012  seq𝜔cseqom 7487  𝑚 cmap 7802  cen 7896  Fincfn 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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709
This theorem is referenced by:  fin23lem32  9110
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