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Theorem fin23lem32 9758
Description: Lemma for fin23 9803. 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 9754 . . . . . . 7 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
87ad2antrl 724 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω–1-1→V)
9 simprl 767 . . . . . . 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 769 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡𝐺)
121, 2, 3, 4, 5, 6fin23lem31 9757 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
139, 10, 11, 12syl3anc 1365 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
14 f1fn 6572 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡 Fn ω)
15 dffn3 6521 . . . . . . . . . . . 12 (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
1614, 15sylib 219 . . . . . . . . . . 11 (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
1716ad2antrl 724 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶ran 𝑡)
18 sspwuni 5018 . . . . . . . . . . . 12 (ran 𝑡 ⊆ 𝒫 𝐺 ran 𝑡𝐺)
1918biimpri 229 . . . . . . . . . . 11 ( ran 𝑡𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
2019ad2antll 725 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
2117, 20fssd 6524 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22 pwexg 5275 . . . . . . . . . . 11 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
2322adantr 481 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝒫 𝐺 ∈ V)
24 vex 3502 . . . . . . . . . . . 12 𝑡 ∈ V
25 f1f 6571 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26 dmfex 7632 . . . . . . . . . . . 12 ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
2724, 25, 26sylancr 587 . . . . . . . . . . 11 (𝑡:ω–1-1→V → ω ∈ V)
2827ad2antrl 724 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ω ∈ V)
2923, 28elmapd 8413 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → (𝑡 ∈ (𝒫 𝐺m ω) ↔ 𝑡:ω⟶𝒫 𝐺))
3021, 29mpbird 258 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡 ∈ (𝒫 𝐺m ω))
31 f1f 6571 . . . . . . . . . 10 (𝑍:ω–1-1→V → 𝑍:ω⟶V)
328, 31syl 17 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω⟶V)
33 fex 6987 . . . . . . . . 9 ((𝑍:ω⟶V ∧ ω ∈ V) → 𝑍 ∈ V)
3432, 28, 33syl2anc 584 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍 ∈ V)
35 eqid 2825 . . . . . . . . 9 (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)
3635fvmpt2 6774 . . . . . . . 8 ((𝑡 ∈ (𝒫 𝐺m ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍)
3730, 34, 36syl2anc 584 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍)
38 f1eq1 6566 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V ↔ 𝑍:ω–1-1→V))
39 rneq 5804 . . . . . . . . . 10 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4039unieqd 4846 . . . . . . . . 9 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4140psseq1d 4072 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) = 𝑍 → ( ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡 ran 𝑍 ran 𝑡))
4238, 41anbi12d 630 . . . . . . 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)))
4337, 42syl 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)))
448, 13, 43mpbir2and 709 . . . . 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))
4544ex 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)))
4645alrimiv 1921 . . 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)))
47 ovex 7184 . . . . 5 (𝒫 𝐺m ω) ∈ V
4847mptex 6984 . . . 4 (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) ∈ V
49 nfmpt1 5160 . . . . . 6 𝑡(𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)
5049nfeq2 2999 . . . . 5 𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)
51 fveq1 6665 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → (𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡))
52 f1eq1 6566 . . . . . . . 8 ((𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) → ((𝑓𝑡):ω–1-1→V ↔ ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V))
5351, 52syl 17 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ((𝑓𝑡):ω–1-1→V ↔ ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡):ω–1-1→V))
5451rneqd 5806 . . . . . . . . 9 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡))
5554unieqd 4846 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡))
5655psseq1d 4072 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍) → ( ran (𝑓𝑡) ⊊ ran 𝑡 ran ((𝑡 ∈ (𝒫 𝐺m ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))
5753, 56anbi12d 630 . . . . . 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)))
5857imbi2d 342 . . . . 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))))
5950, 58albid 2217 . . . 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))))
6048, 59spcev 3610 . . 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)))
6146, 60syl 17 . 2 (𝐺𝐹 → ∃𝑓𝑡((𝑡:ω–1-1→V ∧ ran 𝑡𝐺) → ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
62 f1eq1 6566 . . . . . 6 (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
63 rneq 5804 . . . . . . . 8 (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
6463unieqd 4846 . . . . . . 7 (𝑏 = 𝑡 ran 𝑏 = ran 𝑡)
6564sseq1d 4001 . . . . . 6 (𝑏 = 𝑡 → ( ran 𝑏𝐺 ran 𝑡𝐺))
6662, 65anbi12d 630 . . . . 5 (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) ↔ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)))
67 fveq2 6666 . . . . . . 7 (𝑏 = 𝑡 → (𝑓𝑏) = (𝑓𝑡))
68 f1eq1 6566 . . . . . . 7 ((𝑓𝑏) = (𝑓𝑡) → ((𝑓𝑏):ω–1-1→V ↔ (𝑓𝑡):ω–1-1→V))
6967, 68syl 17 . . . . . 6 (𝑏 = 𝑡 → ((𝑓𝑏):ω–1-1→V ↔ (𝑓𝑡):ω–1-1→V))
7067rneqd 5806 . . . . . . . 8 (𝑏 = 𝑡 → ran (𝑓𝑏) = ran (𝑓𝑡))
7170unieqd 4846 . . . . . . 7 (𝑏 = 𝑡 ran (𝑓𝑏) = ran (𝑓𝑡))
7271, 64psseq12d 4074 . . . . . 6 (𝑏 = 𝑡 → ( ran (𝑓𝑏) ⊊ ran 𝑏 ran (𝑓𝑡) ⊊ ran 𝑡))
7369, 72anbi12d 630 . . . . 5 (𝑏 = 𝑡 → (((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏) ↔ ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
7466, 73imbi12d 346 . . . 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))))
7574cbvalvw 2036 . . 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)))
7675exbii 1841 . 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)))
7761, 76sylibr 235 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 207  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107  {cab 2803  wral 3142  {crab 3146  Vcvv 3499  cdif 3936  cin 3938  wss 3939  wpss 3940  c0 4294  ifcif 4469  𝒫 cpw 4541   cuni 4836   cint 4873   class class class wbr 5062  cmpt 5142  ran crn 5554  ccom 5557  suc csuc 6190   Fn wfn 6346  wf 6347  1-1wf1 6348  cfv 6351  crio 7108  (class class class)co 7151  cmpo 7153  ωcom 7571  seqωcseqom 8077  m cmap 8399  cen 8498  Fincfn 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360
This theorem is referenced by:  fin23lem33  9759
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