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Theorem fin23lem32 10280
Description: Lemma for fin23 10325. Wrap the previous construction into a function to hide the hypotheses. (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
fin23lem32 (𝐺𝐹 → ∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)))
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧   𝑎,𝑏,𝑖,𝑢,𝑡   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃,𝑏   𝑣,𝑎,𝑅,𝑏,𝑖,𝑢   𝑈,𝑎,𝑏,𝑖,𝑢,𝑣,𝑧   𝑓,𝑎,𝑍,𝑏   𝑔,𝑎,𝐺,𝑏,𝑡,𝑓,𝑥
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑓,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,𝑖,𝑎,𝑏)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑓,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑓,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑖,𝑏)   𝐺(𝑧,𝑤,𝑣,𝑢,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem32
StepHypRef Expression
1 fin23lem.a . . . . . . . 8 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
2 fin23lem17.f . . . . . . . 8 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
3 fin23lem.b . . . . . . . 8 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
4 fin23lem.c . . . . . . . 8 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
5 fin23lem.d . . . . . . . 8 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
6 fin23lem.e . . . . . . . 8 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
71, 2, 3, 4, 5, 6fin23lem28 10276 . . . . . . 7 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
87ad2antrl 726 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω–1-1→V)
9 simprl 769 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω–1-1→V)
10 simpl 483 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝐺𝐹)
11 simprr 771 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡𝐺)
121, 2, 3, 4, 5, 6fin23lem31 10279 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
139, 10, 11, 12syl3anc 1371 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
14 f1fn 6739 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡 Fn ω)
15 dffn3 6681 . . . . . . . . . . . 12 (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
1614, 15sylib 217 . . . . . . . . . . 11 (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
1716ad2antrl 726 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶ran 𝑡)
18 sspwuni 5060 . . . . . . . . . . . 12 (ran 𝑡 ⊆ 𝒫 𝐺 ran 𝑡𝐺)
1918biimpri 227 . . . . . . . . . . 11 ( ran 𝑡𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
2019ad2antll 727 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
2117, 20fssd 6686 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22 pwexg 5333 . . . . . . . . . . 11 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
2322adantr 481 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝒫 𝐺 ∈ V)
24 vex 3449 . . . . . . . . . . . 12 𝑡 ∈ V
25 f1f 6738 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26 dmfex 7844 . . . . . . . . . . . 12 ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
2724, 25, 26sylancr 587 . . . . . . . . . . 11 (𝑡:ω–1-1→V → ω ∈ V)
2827ad2antrl 726 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ω ∈ V)
2923, 28elmapd 8779 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → (𝑡 ∈ (𝒫 𝐺m ω) ↔ 𝑡:ω⟶𝒫 𝐺))
3021, 29mpbird 256 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡 ∈ (𝒫 𝐺m ω))
31 f1f 6738 . . . . . . . . . 10 (𝑍:ω–1-1→V → 𝑍:ω⟶V)
328, 31syl 17 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω⟶V)
3332, 28fexd 7177 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍 ∈ V)
34 eqid 2736 . . . . . . . . 9 (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)
3534fvmpt2 6959 . . . . . . . 8 ((𝑡 ∈ (𝒫 𝐺m ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍)
3630, 33, 35syl2anc 584 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍)
37 f1eq1 6733 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ↔ 𝑍:ω–1-1→V))
38 rneq 5891 . . . . . . . . . 10 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
3938unieqd 4879 . . . . . . . . 9 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4039psseq1d 4052 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ( ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡 ran 𝑍 ran 𝑡))
4137, 40anbi12d 631 . . . . . . 7 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ((((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡) ↔ (𝑍:ω–1-1→V ∧ ran 𝑍 ran 𝑡)))
4236, 41syl 17 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ((((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡) ↔ (𝑍:ω–1-1→V ∧ ran 𝑍 ran 𝑡)))
438, 13, 42mpbir2and 711 . . . . 5 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))
4443ex 413 . . . 4 (𝐺𝐹 → ((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡)))
4544alrimiv 1930 . . 3 (𝐺𝐹 → ∀𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡)))
46 ovex 7390 . . . . 5 (𝒫 𝐺m ω) ∈ V
4746mptex 7173 . . . 4 (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) ∈ V
48 nfmpt1 5213 . . . . . 6 𝑡(𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)
4948nfeq2 2924 . . . . 5 𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)
50 fveq1 6841 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → (𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡))
51 f1eq1 6733 . . . . . . . 8 ((𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) → ((𝑓𝑡):ω–1-1→V ↔ ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V))
5250, 51syl 17 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ((𝑓𝑡):ω–1-1→V ↔ ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V))
5350rneqd 5893 . . . . . . . . 9 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡))
5453unieqd 4879 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡))
5554psseq1d 4052 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ( ran (𝑓𝑡) ⊊ ran 𝑡 ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))
5652, 55anbi12d 631 . . . . . 6 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → (((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡) ↔ (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡)))
5756imbi2d 340 . . . . 5 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → (((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)) ↔ ((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))))
5849, 57albid 2215 . . . 4 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → (∀𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)) ↔ ∀𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))))
5947, 58spcev 3565 . . 3 (∀𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ∧ ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡)) → ∃𝑓𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
6045, 59syl 17 . 2 (𝐺𝐹 → ∃𝑓𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
61 f1eq1 6733 . . . . . 6 (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
62 rneq 5891 . . . . . . . 8 (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
6362unieqd 4879 . . . . . . 7 (𝑏 = 𝑡 ran 𝑏 = ran 𝑡)
6463sseq1d 3975 . . . . . 6 (𝑏 = 𝑡 → ( ran 𝑏𝐺 ran 𝑡𝐺))
6561, 64anbi12d 631 . . . . 5 (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) ↔ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)))
66 fveq2 6842 . . . . . . 7 (𝑏 = 𝑡 → (𝑓𝑏) = (𝑓𝑡))
67 f1eq1 6733 . . . . . . 7 ((𝑓𝑏) = (𝑓𝑡) → ((𝑓𝑏):ω–1-1→V ↔ (𝑓𝑡):ω–1-1→V))
6866, 67syl 17 . . . . . 6 (𝑏 = 𝑡 → ((𝑓𝑏):ω–1-1→V ↔ (𝑓𝑡):ω–1-1→V))
6966rneqd 5893 . . . . . . . 8 (𝑏 = 𝑡 → ran (𝑓𝑏) = ran (𝑓𝑡))
7069unieqd 4879 . . . . . . 7 (𝑏 = 𝑡 ran (𝑓𝑏) = ran (𝑓𝑡))
7170, 63psseq12d 4054 . . . . . 6 (𝑏 = 𝑡 → ( ran (𝑓𝑏) ⊊ ran 𝑏 ran (𝑓𝑡) ⊊ ran 𝑡))
7268, 71anbi12d 631 . . . . 5 (𝑏 = 𝑡 → (((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏) ↔ ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
7365, 72imbi12d 344 . . . 4 (𝑏 = 𝑡 → (((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)) ↔ ((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡))))
7473cbvalvw 2039 . . 3 (∀𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)) ↔ ∀𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
7574exbii 1850 . 2 (∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)) ↔ ∃𝑓𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
7660, 75sylibr 233 1 (𝐺𝐹 → ∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  cin 3909  wss 3910  wpss 3911  c0 4282  ifcif 4486  𝒫 cpw 4560   cuni 4865   cint 4907   class class class wbr 5105  cmpt 5188  ran crn 5634  ccom 5637  suc csuc 6319   Fn wfn 6491  wf 6492  1-1wf1 6493  cfv 6496  crio 7312  (class class class)co 7357  cmpo 7359  ωcom 7802  seqωcseqom 8393  m cmap 8765  cen 8880  Fincfn 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seqom 8394  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875
This theorem is referenced by:  fin23lem33  10281
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