MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem32 Structured version   Visualization version   GIF version

Theorem fin23lem32 9117
Description: Lemma for fin23 9162. 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 9113 . . . . . . 7 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
87ad2antrl 763 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍:ω–1-1→V)
9 simprl 793 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω–1-1→V)
10 simpl 473 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝐺𝐹)
11 simprr 795 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡𝐺)
121, 2, 3, 4, 5, 6fin23lem31 9116 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
139, 10, 11, 12syl3anc 1323 . . . . . 6 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
14 f1fn 6064 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡 Fn ω)
15 dffn3 6016 . . . . . . . . . . . 12 (𝑡 Fn ω ↔ 𝑡:ω⟶ran 𝑡)
1614, 15sylib 208 . . . . . . . . . . 11 (𝑡:ω–1-1→V → 𝑡:ω⟶ran 𝑡)
1716ad2antrl 763 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶ran 𝑡)
18 sspwuni 4582 . . . . . . . . . . . 12 (ran 𝑡 ⊆ 𝒫 𝐺 ran 𝑡𝐺)
1918biimpri 218 . . . . . . . . . . 11 ( ran 𝑡𝐺 → ran 𝑡 ⊆ 𝒫 𝐺)
2019ad2antll 764 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ran 𝑡 ⊆ 𝒫 𝐺)
2117, 20fssd 6019 . . . . . . . . 9 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑡:ω⟶𝒫 𝐺)
22 pwexg 4815 . . . . . . . . . . 11 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
2322adantr 481 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝒫 𝐺 ∈ V)
24 vex 3192 . . . . . . . . . . . 12 𝑡 ∈ V
25 f1f 6063 . . . . . . . . . . . 12 (𝑡:ω–1-1→V → 𝑡:ω⟶V)
26 dmfex 7078 . . . . . . . . . . . 12 ((𝑡 ∈ V ∧ 𝑡:ω⟶V) → ω ∈ V)
2724, 25, 26sylancr 694 . . . . . . . . . . 11 (𝑡:ω–1-1→V → ω ∈ V)
2827ad2antrl 763 . . . . . . . . . 10 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ω ∈ V)
2923, 28elmapd 7823 . . . . . . . . 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 6063 . . . . . . . . . 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 6450 . . . . . . . . 9 ((𝑍:ω⟶V ∧ ω ∈ V) → 𝑍 ∈ V)
3432, 28, 33syl2anc 692 . . . . . . . 8 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → 𝑍 ∈ V)
35 eqid 2621 . . . . . . . . 9 (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
3635fvmpt2 6253 . . . . . . . 8 ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ∧ 𝑍 ∈ V) → ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍)
3730, 34, 36syl2anc 692 . . . . . . 7 ((𝐺𝐹 ∧ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)) → ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍)
38 f1eq1 6058 . . . . . . . 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 5316 . . . . . . . . . 10 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 → ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4039unieqd 4417 . . . . . . . . 9 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = ran 𝑍)
4140psseq1d 3682 . . . . . . . 8 (((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) = 𝑍 → ( ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡 ran 𝑍 ran 𝑡))
4238, 41anbi12d 746 . . . . . . 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 956 . . . . 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 450 . . . 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 1852 . . 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 6638 . . . . 5 (𝒫 𝐺𝑚 ω) ∈ V
4847mptex 6446 . . . 4 (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) ∈ V
49 nfmpt1 4712 . . . . . 6 𝑡(𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
5049nfeq2 2776 . . . . 5 𝑡 𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)
51 fveq1 6152 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → (𝑓𝑡) = ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
52 f1eq1 6058 . . . . . . . 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 5318 . . . . . . . . 9 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
5554unieqd 4417 . . . . . . . 8 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ran (𝑓𝑡) = ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡))
5655psseq1d 3682 . . . . . . 7 (𝑓 = (𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍) → ( ran (𝑓𝑡) ⊊ ran 𝑡 ran ((𝑡 ∈ (𝒫 𝐺𝑚 ω) ↦ 𝑍)‘𝑡) ⊊ ran 𝑡))
5753, 56anbi12d 746 . . . . . 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 330 . . . . 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 2088 . . . 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 3289 . . 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 6058 . . . . . 6 (𝑏 = 𝑡 → (𝑏:ω–1-1→V ↔ 𝑡:ω–1-1→V))
63 rneq 5316 . . . . . . . 8 (𝑏 = 𝑡 → ran 𝑏 = ran 𝑡)
6463unieqd 4417 . . . . . . 7 (𝑏 = 𝑡 ran 𝑏 = ran 𝑡)
6564sseq1d 3616 . . . . . 6 (𝑏 = 𝑡 → ( ran 𝑏𝐺 ran 𝑡𝐺))
6662, 65anbi12d 746 . . . . 5 (𝑏 = 𝑡 → ((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) ↔ (𝑡:ω–1-1→V ∧ ran 𝑡𝐺)))
67 fveq2 6153 . . . . . . 7 (𝑏 = 𝑡 → (𝑓𝑏) = (𝑓𝑡))
68 f1eq1 6058 . . . . . . 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 5318 . . . . . . . 8 (𝑏 = 𝑡 → ran (𝑓𝑏) = ran (𝑓𝑡))
7170unieqd 4417 . . . . . . 7 (𝑏 = 𝑡 ran (𝑓𝑏) = ran (𝑓𝑡))
7271, 64psseq12d 3684 . . . . . 6 (𝑏 = 𝑡 → ( ran (𝑓𝑏) ⊊ ran 𝑏 ran (𝑓𝑡) ⊊ ran 𝑡))
7369, 72anbi12d 746 . . . . 5 (𝑏 = 𝑡 → (((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏) ↔ ((𝑓𝑡):ω–1-1→V ∧ ran (𝑓𝑡) ⊊ ran 𝑡)))
7466, 73imbi12d 334 . . . 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 2272 . . 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 1771 . 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 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  {crab 2911  Vcvv 3189  cdif 3556  cin 3558  wss 3559  wpss 3560  c0 3896  ifcif 4063  𝒫 cpw 4135   cuni 4407   cint 4445   class class class wbr 4618  cmpt 4678  ran crn 5080  ccom 5083  suc csuc 5689   Fn wfn 5847  wf 5848  1-1wf1 5849  cfv 5852  crio 6570  (class class class)co 6610  cmpt2 6612  ωcom 7019  seq𝜔cseqom 7494  𝑚 cmap 7809  cen 7903  Fincfn 7906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-seqom 7495  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716
This theorem is referenced by:  fin23lem33  9118
  Copyright terms: Public domain W3C validator