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Theorem fin23lem32 9368
Description: Lemma for fin23 9413. 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 9364 . . . . . . 7 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
87ad2antrl 699 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω–1-1→V)
9 simprl 746 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω–1-1→V)
10 simpl 468 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝐺𝐹)
11 simprr 748 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡𝐺)
121, 2, 3, 4, 5, 6fin23lem31 9367 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
139, 10, 11, 12syl3anc 1476 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
14 f1fn 6242 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡 Fn ω)
15 dffn3 6194 . . . . . . . . . . . 12 (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
1614, 15sylib 208 . . . . . . . . . . 11 (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
1716ad2antrl 699 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶ran 𝑡)
18 sspwuni 4745 . . . . . . . . . . . 12 (ran 𝑡 ⊆ 𝒫 𝐺 ran 𝑡𝐺)
1918biimpri 218 . . . . . . . . . . 11 ( ran 𝑡𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
2019ad2antll 700 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
2117, 20fssd 6197 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22 pwexg 4981 . . . . . . . . . . 11 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
2322adantr 466 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝒫 𝐺 ∈ V)
24 vex 3354 . . . . . . . . . . . 12 𝑡 ∈ V
25 f1f 6241 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26 dmfex 7271 . . . . . . . . . . . 12 ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
2724, 25, 26sylancr 567 . . . . . . . . . . 11 (𝑡:ω–1-1→V → ω ∈ V)
2827ad2antrl 699 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ω ∈ V)
2923, 28elmapd 8023 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↔ 𝑡:ω⟶𝒫 𝐺))
3021, 29mpbird 247 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡 ∈ (𝒫 𝐺𝑚 ω))
31 f1f 6241 . . . . . . . . . 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 6633 . . . . . . . . 9 ((𝑍:ω⟶V ∧ ω ∈ V) → 𝑍 ∈ V)
3432, 28, 33syl2anc 565 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍 ∈ V)
35 eqid 2771 . . . . . . . . 9 (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
3635fvmpt2 6433 . . . . . . . 8 ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍)
3730, 34, 36syl2anc 565 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍)
38 f1eq1 6236 . . . . . . . 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 5489 . . . . . . . . . 10 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4039unieqd 4584 . . . . . . . . 9 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4140psseq1d 3849 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 → ( ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡 ran 𝑍 ran 𝑡))
4238, 41anbi12d 608 . . . . . . 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 684 . . . . 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 397 . . . 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 2007 . . 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 6823 . . . . 5 (𝒫 𝐺𝑚 ω) ∈ V
4847mptex 6630 . . . 4 (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) ∈ V
49 nfmpt1 4881 . . . . . 6 𝑡(𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
5049nfeq2 2929 . . . . 5 𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
51 fveq1 6331 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → (𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
52 f1eq1 6236 . . . . . . . 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 5491 . . . . . . . . 9 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
5554unieqd 4584 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
5655psseq1d 3849 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ( ran (𝑓𝑡) ⊊ ran 𝑡 ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))
5753, 56anbi12d 608 . . . . . 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 329 . . . . 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 2246 . . . 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 3451 . . 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 6236 . . . . . 6 (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
63 rneq 5489 . . . . . . . 8 (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
6463unieqd 4584 . . . . . . 7 (𝑏 = 𝑡 ran 𝑏 = ran 𝑡)
6564sseq1d 3781 . . . . . 6 (𝑏 = 𝑡 → ( ran 𝑏𝐺 ran 𝑡𝐺))
6662, 65anbi12d 608 . . . . 5 (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) ↔ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)))
67 fveq2 6332 . . . . . . 7 (𝑏 = 𝑡 → (𝑓𝑏) = (𝑓𝑡))
68 f1eq1 6236 . . . . . . 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 5491 . . . . . . . 8 (𝑏 = 𝑡 → ran (𝑓𝑏) = ran (𝑓𝑡))
7170unieqd 4584 . . . . . . 7 (𝑏 = 𝑡 ran (𝑓𝑏) = ran (𝑓𝑡))
7271, 64psseq12d 3851 . . . . . 6 (𝑏 = 𝑡 → ( ran (𝑓𝑏) ⊊ ran 𝑏 ran (𝑓𝑡) ⊊ ran 𝑡))
7369, 72anbi12d 608 . . . . 5 (𝑏 = 𝑡 → (((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏) ↔ ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
7466, 73imbi12d 333 . . . 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 2125 . . 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 1924 . 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 224 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 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  {crab 3065  Vcvv 3351  cdif 3720  cin 3722  wss 3723  wpss 3724  c0 4063  ifcif 4225  𝒫 cpw 4297   cuni 4574   cint 4611   class class class wbr 4786  cmpt 4863  ran crn 5250  ccom 5253  suc csuc 5868   Fn wfn 6026  wf 6027  1-1wf1 6028  cfv 6031  crio 6753  (class class class)co 6793  cmpt2 6795  ωcom 7212  seq𝜔cseqom 7695  𝑚 cmap 8009  cen 8106  Fincfn 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965
This theorem is referenced by:  fin23lem33  9369
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