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Theorem fin1a2lem5 10444
Description: Lemma for fin1a2 10455. (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 8694 . 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 10443 . . . . 5 𝐸:ω–1-1→ω
4 f1fn 6805 . . . . 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 2744 . . . . 5 ((𝐸𝑎) = 𝐴𝐴 = (𝐸𝑎))
92fin1a2lem3 10442 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2o ·o 𝑎))
109eqeq2d 2748 . . . . 5 (𝑎 ∈ ω → (𝐴 = (𝐸𝑎) ↔ 𝐴 = (2o ·o 𝑎)))
118, 10bitrid 283 . . . 4 (𝑎 ∈ ω → ((𝐸𝑎) = 𝐴𝐴 = (2o ·o 𝑎)))
1211rexbiia 3092 . . 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 2744 . . . . . 6 ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸𝑎))
179eqeq2d 2748 . . . . . 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 3092 . . . 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 1540  wcel 2108  wrex 3070  cmpt 5225  ran crn 5686  suc csuc 6386   Fn wfn 6556  1-1wf1 6558  cfv 6561  (class class class)co 7431  ωcom 7887  2oc2o 8500   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511
This theorem is referenced by:  fin1a2lem6  10445
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