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Theorem fin1a2lem5 9427
Description: Lemma for fin1a2 9438. (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 7885 . 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 9426 . . . . 5 𝐸:ω–1-1→ω
4 f1fn 6242 . . . . 5 (𝐸:ω–1-1→ω → 𝐸 Fn ω)
53, 4ax-mp 5 . . . 4 𝐸 Fn ω
6 fvelrnb 6385 . . . 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 2777 . . . . 5 ((𝐸𝑎) = 𝐴𝐴 = (𝐸𝑎))
92fin1a2lem3 9425 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2𝑜 ·𝑜 𝑎))
109eqeq2d 2780 . . . . 5 (𝑎 ∈ ω → (𝐴 = (𝐸𝑎) ↔ 𝐴 = (2𝑜 ·𝑜 𝑎)))
118, 10syl5bb 272 . . . 4 (𝑎 ∈ ω → ((𝐸𝑎) = 𝐴𝐴 = (2𝑜 ·𝑜 𝑎)))
1211rexbiia 3187 . . 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 6385 . . . . 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 2777 . . . . . 6 ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸𝑎))
179eqeq2d 2780 . . . . . 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 3187 . . . 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 1630  wcel 2144  wrex 3061  cmpt 4861  ran crn 5250  suc csuc 5868   Fn wfn 6026  1-1wf1 6028  cfv 6031  (class class class)co 6792  ωcom 7211  2𝑜c2o 7706   ·𝑜 comu 7710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  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-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-omul 7717
This theorem is referenced by:  fin1a2lem6  9428
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