MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem28 Structured version   Visualization version   GIF version

Theorem fin23lem28 9354
Description: Lemma for fin23 9403. 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 4270 . . 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 3880 . . . . . . . . 9 (ω ∖ 𝑃) ⊆ ω
5 ominf 8337 . . . . . . . . . 10 ¬ ω ∈ Fin
6 fin23lem.b . . . . . . . . . . . . . 14 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
7 ssrab2 3828 . . . . . . . . . . . . . 14 {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ⊆ ω
86, 7eqsstri 3776 . . . . . . . . . . . . 13 𝑃 ⊆ ω
9 undif 4193 . . . . . . . . . . . . 13 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
108, 9mpbi 220 . . . . . . . . . . . 12 (𝑃 ∪ (ω ∖ 𝑃)) = ω
11 unfi 8392 . . . . . . . . . . . 12 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
1210, 11syl5eqelr 2844 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
1312ex 449 . . . . . . . . . 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 9341 . . . . . . . . 9 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
174, 14, 16sylancr 698 . . . . . . . 8 (𝑃 ∈ Fin → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
1817adantl 473 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
19 f1of1 6297 . . . . . . 7 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω–1-1→(ω ∖ 𝑃))
20 f1ss 6267 . . . . . . . 8 ((𝑅:ω–1-1→(ω ∖ 𝑃) ∧ (ω ∖ 𝑃) ⊆ ω) → 𝑅:ω–1-1→ω)
214, 20mpan2 709 . . . . . . 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 6271 . . . . . 6 ((𝑡:ω–1-1→V ∧ 𝑅:ω–1-1→ω) → (𝑡𝑅):ω–1-1→V)
2422, 23syldan 488 . . . . 5 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑡𝑅):ω–1-1→V)
25 f1eq1 6257 . . . . 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 650 . . 3 ((𝑡:ω–1-1→V ∧ (𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅))) → 𝑍:ω–1-1→V)
28 fvex 6362 . . . . . . . . . . 11 (𝑡𝑧) ∈ V
29 difexg 4960 . . . . . . . . . . 11 ((𝑡𝑧) ∈ V → ((𝑡𝑧) ∖ ran 𝑈) ∈ V)
3028, 29ax-mp 5 . . . . . . . . . 10 ((𝑡𝑧) ∖ ran 𝑈) ∈ V
3130rgenw 3062 . . . . . . . . 9 𝑧𝑃 ((𝑡𝑧) ∖ ran 𝑈) ∈ V
32 eqid 2760 . . . . . . . . . 10 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
3332fmpt 6544 . . . . . . . . 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 6352 . . . . . . . . . . . . 13 (𝑧 = 𝑎 → (𝑡𝑧) = (𝑡𝑎))
3736difeq1d 3870 . . . . . . . . . . . 12 (𝑧 = 𝑎 → ((𝑡𝑧) ∖ ran 𝑈) = ((𝑡𝑎) ∖ ran 𝑈))
38 fvex 6362 . . . . . . . . . . . . 13 (𝑡𝑎) ∈ V
39 difexg 4960 . . . . . . . . . . . . 13 ((𝑡𝑎) ∈ V → ((𝑡𝑎) ∖ ran 𝑈) ∈ V)
4038, 39ax-mp 5 . . . . . . . . . . . 12 ((𝑡𝑎) ∖ ran 𝑈) ∈ V
4137, 32, 40fvmpt 6444 . . . . . . . . . . 11 (𝑎𝑃 → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑡𝑎) ∖ ran 𝑈))
4241ad2antrl 766 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑡𝑎) ∖ ran 𝑈))
43 fveq2 6352 . . . . . . . . . . . . 13 (𝑧 = 𝑏 → (𝑡𝑧) = (𝑡𝑏))
4443difeq1d 3870 . . . . . . . . . . . 12 (𝑧 = 𝑏 → ((𝑡𝑧) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈))
45 fvex 6362 . . . . . . . . . . . . 13 (𝑡𝑏) ∈ V
46 difexg 4960 . . . . . . . . . . . . 13 ((𝑡𝑏) ∈ V → ((𝑡𝑏) ∖ ran 𝑈) ∈ V)
4745, 46ax-mp 5 . . . . . . . . . . . 12 ((𝑡𝑏) ∖ ran 𝑈) ∈ V
4844, 32, 47fvmpt 6444 . . . . . . . . . . 11 (𝑏𝑃 → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) = ((𝑡𝑏) ∖ ran 𝑈))
4948ad2antll 767 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) = ((𝑡𝑏) ∖ ran 𝑈))
5042, 49eqeq12d 2775 . . . . . . . . 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 3904 . . . . . . . . . . 11 (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)))
52 fveq2 6352 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑎 → (𝑡𝑣) = (𝑡𝑎))
5352sseq2d 3774 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑎 → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡𝑎)))
5453, 6elrab2 3507 . . . . . . . . . . . . . . 15 (𝑎𝑃 ↔ (𝑎 ∈ ω ∧ ran 𝑈 ⊆ (𝑡𝑎)))
5554simprbi 483 . . . . . . . . . . . . . 14 (𝑎𝑃 ran 𝑈 ⊆ (𝑡𝑎))
5655ad2antrl 766 . . . . . . . . . . . . 13 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ran 𝑈 ⊆ (𝑡𝑎))
57 undif 4193 . . . . . . . . . . . . 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 6352 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑏 → (𝑡𝑣) = (𝑡𝑏))
6059sseq2d 3774 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑏 → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡𝑏)))
6160, 6elrab2 3507 . . . . . . . . . . . . . . 15 (𝑏𝑃 ↔ (𝑏 ∈ ω ∧ ran 𝑈 ⊆ (𝑡𝑏)))
6261simprbi 483 . . . . . . . . . . . . . 14 (𝑏𝑃 ran 𝑈 ⊆ (𝑡𝑏))
6362ad2antll 767 . . . . . . . . . . . . 13 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ran 𝑈 ⊆ (𝑡𝑏))
64 undif 4193 . . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 3740 . . . . . . . . . . . 12 (𝑎𝑃𝑎 ∈ ω)
698sseli 3740 . . . . . . . . . . . 12 (𝑏𝑃𝑏 ∈ ω)
7068, 69anim12i 591 . . . . . . . . . . 11 ((𝑎𝑃𝑏𝑃) → (𝑎 ∈ ω ∧ 𝑏 ∈ ω))
71 f1fveq 6682 . . . . . . . . . . 11 ((𝑡:ω–1-1→V ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝑡𝑎) = (𝑡𝑏) ↔ 𝑎 = 𝑏))
7270, 71sylan2 492 . . . . . . . . . 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 3109 . . . . . . 7 (𝑡:ω–1-1→V → ∀𝑎𝑃𝑏𝑃 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
76 dff13 6675 . . . . . . 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 701 . . . . . 6 (𝑡:ω–1-1→V → (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V)
78 fin23lem.c . . . . . . . . 9 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
7978fin23lem22 9341 . . . . . . . 8 ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1-onto𝑃)
80 f1of1 6297 . . . . . . . 8 (𝑄:ω–1-1-onto𝑃𝑄:ω–1-1𝑃)
8179, 80syl 17 . . . . . . 7 ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1𝑃)
828, 81mpan 708 . . . . . 6 𝑃 ∈ Fin → 𝑄:ω–1-1𝑃)
83 f1co 6271 . . . . . 6 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V ∧ 𝑄:ω–1-1𝑃) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
8477, 82, 83syl2an 495 . . . . 5 ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
85 f1eq1 6257 . . . . 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 650 . . 3 ((𝑡:ω–1-1→V ∧ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → 𝑍:ω–1-1→V)
8827, 87jaodan 861 . 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 709 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 382  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  ifcif 4230  𝒫 cpw 4302   cuni 4588   cint 4627   class class class wbr 4804  cmpt 4881  ran crn 5267  ccom 5270  suc csuc 5886  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049  crio 6773  (class class class)co 6813  cmpt2 6815  ωcom 7230  seq𝜔cseqom 7711  𝑚 cmap 8023  cen 8118  Fincfn 8121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955
This theorem is referenced by:  fin23lem32  9358
  Copyright terms: Public domain W3C validator