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Theorem fin1a2lem5 10442
Description: Lemma for fin1a2 10453. (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 8693 . 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 10441 . . . . 5 𝐸:ω–1-1→ω
4 f1fn 6806 . . . . 5 (𝐸:ω–1-1→ω → 𝐸 Fn ω)
53, 4ax-mp 5 . . . 4 𝐸 Fn ω
6 fvelrnb 6969 . . . 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 2742 . . . . 5 ((𝐸𝑎) = 𝐴𝐴 = (𝐸𝑎))
92fin1a2lem3 10440 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2o ·o 𝑎))
109eqeq2d 2746 . . . . 5 (𝑎 ∈ ω → (𝐴 = (𝐸𝑎) ↔ 𝐴 = (2o ·o 𝑎)))
118, 10bitrid 283 . . . 4 (𝑎 ∈ ω → ((𝐸𝑎) = 𝐴𝐴 = (2o ·o 𝑎)))
1211rexbiia 3090 . . 3 (∃𝑎 ∈ ω (𝐸𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
137, 12bitri 275 . 2 (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
14 fvelrnb 6969 . . . . 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 2742 . . . . . 6 ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸𝑎))
179eqeq2d 2746 . . . . . 6 (𝑎 ∈ ω → (suc 𝐴 = (𝐸𝑎) ↔ suc 𝐴 = (2o ·o 𝑎)))
1816, 17bitrid 283 . . . . 5 (𝑎 ∈ ω → ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (2o ·o 𝑎)))
1918rexbiia 3090 . . . 4 (∃𝑎 ∈ ω (𝐸𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
2015, 19bitri 275 . . 3 (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
2120notbii 320 . 2 (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
221, 13, 213bitr4g 314 1 (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  wrex 3068  cmpt 5231  ran crn 5690  suc csuc 6388   Fn wfn 6558  1-1wf1 6560  cfv 6563  (class class class)co 7431  ωcom 7887  2oc2o 8499   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510
This theorem is referenced by:  fin1a2lem6  10443
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