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Theorem fin23lem32 9379
Description: Lemma for fin23 9424. 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 9375 . . . . . . 7 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
87ad2antrl 766 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω–1-1→V)
9 simprl 811 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω–1-1→V)
10 simpl 474 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝐺𝐹)
11 simprr 813 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡𝐺)
121, 2, 3, 4, 5, 6fin23lem31 9378 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
139, 10, 11, 12syl3anc 1477 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
14 f1fn 6264 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡 Fn ω)
15 dffn3 6216 . . . . . . . . . . . 12 (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
1614, 15sylib 208 . . . . . . . . . . 11 (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
1716ad2antrl 766 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶ran 𝑡)
18 sspwuni 4764 . . . . . . . . . . . 12 (ran 𝑡 ⊆ 𝒫 𝐺 ran 𝑡𝐺)
1918biimpri 218 . . . . . . . . . . 11 ( ran 𝑡𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
2019ad2antll 767 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
2117, 20fssd 6219 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22 pwexg 5000 . . . . . . . . . . 11 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
2322adantr 472 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝒫 𝐺 ∈ V)
24 vex 3344 . . . . . . . . . . . 12 𝑡 ∈ V
25 f1f 6263 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26 dmfex 7291 . . . . . . . . . . . 12 ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
2724, 25, 26sylancr 698 . . . . . . . . . . 11 (𝑡:ω–1-1→V → ω ∈ V)
2827ad2antrl 766 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ω ∈ V)
2923, 28elmapd 8040 . . . . . . . . 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 6263 . . . . . . . . . 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 6655 . . . . . . . . 9 ((𝑍:ω⟶V ∧ ω ∈ V) → 𝑍 ∈ V)
3432, 28, 33syl2anc 696 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍 ∈ V)
35 eqid 2761 . . . . . . . . 9 (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
3635fvmpt2 6455 . . . . . . . 8 ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍)
3730, 34, 36syl2anc 696 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍)
38 f1eq1 6258 . . . . . . . 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 5507 . . . . . . . . . 10 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4039unieqd 4599 . . . . . . . . 9 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4140psseq1d 3842 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 → ( ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡 ran 𝑍 ran 𝑡))
4238, 41anbi12d 749 . . . . . . 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 995 . . . . 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 449 . . . 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 2005 . . 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 6843 . . . . 5 (𝒫 𝐺𝑚 ω) ∈ V
4847mptex 6652 . . . 4 (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) ∈ V
49 nfmpt1 4900 . . . . . 6 𝑡(𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
5049nfeq2 2919 . . . . 5 𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
51 fveq1 6353 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → (𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
52 f1eq1 6258 . . . . . . . 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 5509 . . . . . . . . 9 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
5554unieqd 4599 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
5655psseq1d 3842 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ( ran (𝑓𝑡) ⊊ ran 𝑡 ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))
5753, 56anbi12d 749 . . . . . 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 2238 . . . 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 3441 . . 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 6258 . . . . . 6 (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
63 rneq 5507 . . . . . . . 8 (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
6463unieqd 4599 . . . . . . 7 (𝑏 = 𝑡 ran 𝑏 = ran 𝑡)
6564sseq1d 3774 . . . . . 6 (𝑏 = 𝑡 → ( ran 𝑏𝐺 ran 𝑡𝐺))
6662, 65anbi12d 749 . . . . 5 (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) ↔ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)))
67 fveq2 6354 . . . . . . 7 (𝑏 = 𝑡 → (𝑓𝑏) = (𝑓𝑡))
68 f1eq1 6258 . . . . . . 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 5509 . . . . . . . 8 (𝑏 = 𝑡 → ran (𝑓𝑏) = ran (𝑓𝑡))
7170unieqd 4599 . . . . . . 7 (𝑏 = 𝑡 ran (𝑓𝑏) = ran (𝑓𝑡))
7271, 64psseq12d 3844 . . . . . 6 (𝑏 = 𝑡 → ( ran (𝑓𝑏) ⊊ ran 𝑏 ran (𝑓𝑡) ⊊ ran 𝑡))
7369, 72anbi12d 749 . . . . 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))))
7574cbvalv 2419 . . 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 1923 . 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 383  wal 1630   = wceq 1632  wex 1853  wcel 2140  {cab 2747  wral 3051  {crab 3055  Vcvv 3341  cdif 3713  cin 3715  wss 3716  wpss 3717  c0 4059  ifcif 4231  𝒫 cpw 4303   cuni 4589   cint 4628   class class class wbr 4805  cmpt 4882  ran crn 5268  ccom 5271  suc csuc 5887   Fn wfn 6045  wf 6046  1-1wf1 6047  cfv 6050  crio 6775  (class class class)co 6815  cmpt2 6817  ωcom 7232  seq𝜔cseqom 7713  𝑚 cmap 8026  cen 8121  Fincfn 8124
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-seqom 7714  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-card 8976
This theorem is referenced by:  fin23lem33  9380
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