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Theorem fin1a2lem5 9264
Description: Lemma for fin1a2 9275. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.b 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
Assertion
Ref Expression
fin1a2lem5 (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))

Proof of Theorem fin1a2lem5
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 nneob 7777 . 2 (𝐴 ∈ ω → (∃𝑎 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑎) ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎)))
2 fin1a2lem.b . . . . . 6 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
32fin1a2lem4 9263 . . . . 5 𝐸:ω–1-1→ω
4 f1fn 6140 . . . . 5 (𝐸:ω–1-1→ω → 𝐸 Fn ω)
53, 4ax-mp 5 . . . 4 𝐸 Fn ω
6 fvelrnb 6282 . . . 4 (𝐸 Fn ω → (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸𝑎) = 𝐴))
75, 6ax-mp 5 . . 3 (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸𝑎) = 𝐴)
8 eqcom 2658 . . . . 5 ((𝐸𝑎) = 𝐴𝐴 = (𝐸𝑎))
92fin1a2lem3 9262 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2𝑜 ·𝑜 𝑎))
109eqeq2d 2661 . . . . 5 (𝑎 ∈ ω → (𝐴 = (𝐸𝑎) ↔ 𝐴 = (2𝑜 ·𝑜 𝑎)))
118, 10syl5bb 272 . . . 4 (𝑎 ∈ ω → ((𝐸𝑎) = 𝐴𝐴 = (2𝑜 ·𝑜 𝑎)))
1211rexbiia 3069 . . 3 (∃𝑎 ∈ ω (𝐸𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑎))
137, 12bitri 264 . 2 (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2𝑜 ·𝑜 𝑎))
14 fvelrnb 6282 . . . . 5 (𝐸 Fn ω → (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸𝑎) = suc 𝐴))
155, 14ax-mp 5 . . . 4 (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸𝑎) = suc 𝐴)
16 eqcom 2658 . . . . . 6 ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸𝑎))
179eqeq2d 2661 . . . . . 6 (𝑎 ∈ ω → (suc 𝐴 = (𝐸𝑎) ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑎)))
1816, 17syl5bb 272 . . . . 5 (𝑎 ∈ ω → ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (2𝑜 ·𝑜 𝑎)))
1918rexbiia 3069 . . . 4 (∃𝑎 ∈ ω (𝐸𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎))
2015, 19bitri 264 . . 3 (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎))
2120notbii 309 . 2 (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2𝑜 ·𝑜 𝑎))
221, 13, 213bitr4g 303 1 (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1523  wcel 2030  wrex 2942  cmpt 4762  ran crn 5144  suc csuc 5763   Fn wfn 5921  1-1wf1 5923  cfv 5926  (class class class)co 6690  ωcom 7107  2𝑜c2o 7599   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610
This theorem is referenced by:  fin1a2lem6  9265
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