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Theorem fin1a2lem4 9409
Description: Lemma for fin1a2 9421. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.b 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
Assertion
Ref Expression
fin1a2lem4 𝐸:ω–1-1→ω

Proof of Theorem fin1a2lem4
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
2 2onn 7881 . . . 4 2𝑜 ∈ ω
3 nnmcl 7853 . . . 4 ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω)
42, 3mpan 708 . . 3 (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω)
51, 4fmpti 6538 . 2 𝐸:ω⟶ω
61fin1a2lem3 9408 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2𝑜 ·𝑜 𝑎))
71fin1a2lem3 9408 . . . . . 6 (𝑏 ∈ ω → (𝐸𝑏) = (2𝑜 ·𝑜 𝑏))
86, 7eqeqan12d 2768 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏)))
9 2on 7729 . . . . . . 7 2𝑜 ∈ On
109a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On)
11 nnon 7228 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ On)
1211adantr 472 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13 nnon 7228 . . . . . . 7 (𝑏 ∈ ω → 𝑏 ∈ On)
1413adantl 473 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15 0lt1o 7745 . . . . . . . . 9 ∅ ∈ 1𝑜
16 elelsuc 5950 . . . . . . . . 9 (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜)
1715, 16ax-mp 5 . . . . . . . 8 ∅ ∈ suc 1𝑜
18 df-2o 7722 . . . . . . . 8 2𝑜 = suc 1𝑜
1917, 18eleqtrri 2830 . . . . . . 7 ∅ ∈ 2𝑜
2019a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜)
21 omcan 7810 . . . . . 6 (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
2210, 12, 14, 20, 21syl31anc 1476 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
238, 22bitrd 268 . . . 4 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ 𝑎 = 𝑏))
2423biimpd 219 . . 3 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏))
2524rgen2a 3107 . 2 𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)
26 dff13 6667 . 2 (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)))
275, 25, 26mpbir2an 993 1 𝐸:ω–1-1→ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  c0 4050  cmpt 4873  Oncon0 5876  suc csuc 5878  wf 6037  1-1wf1 6038  cfv 6041  (class class class)co 6805  ωcom 7222  1𝑜c1o 7714  2𝑜c2o 7715   ·𝑜 comu 7719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-omul 7726
This theorem is referenced by:  fin1a2lem5  9410  fin1a2lem6  9411  fin1a2lem7  9412
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