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Theorem fin1a2lem5 9803
Description: Lemma for fin1a2 9814. (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 8254 . 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 9802 . . . . 5 𝐸:ω–1-1→ω
4 f1fn 6549 . . . . 5 (𝐸:ω–1-1→ω → 𝐸 Fn ω)
53, 4ax-mp 5 . . . 4 𝐸 Fn ω
6 fvelrnb 6699 . . . 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 2828 . . . . 5 ((𝐸𝑎) = 𝐴𝐴 = (𝐸𝑎))
92fin1a2lem3 9801 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2o ·o 𝑎))
109eqeq2d 2832 . . . . 5 (𝑎 ∈ ω → (𝐴 = (𝐸𝑎) ↔ 𝐴 = (2o ·o 𝑎)))
118, 10syl5bb 286 . . . 4 (𝑎 ∈ ω → ((𝐸𝑎) = 𝐴𝐴 = (2o ·o 𝑎)))
1211rexbiia 3234 . . 3 (∃𝑎 ∈ ω (𝐸𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
137, 12bitri 278 . 2 (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎))
14 fvelrnb 6699 . . . . 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 2828 . . . . . 6 ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸𝑎))
179eqeq2d 2832 . . . . . 6 (𝑎 ∈ ω → (suc 𝐴 = (𝐸𝑎) ↔ suc 𝐴 = (2o ·o 𝑎)))
1816, 17syl5bb 286 . . . . 5 (𝑎 ∈ ω → ((𝐸𝑎) = suc 𝐴 ↔ suc 𝐴 = (2o ·o 𝑎)))
1918rexbiia 3234 . . . 4 (∃𝑎 ∈ ω (𝐸𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
2015, 19bitri 278 . . 3 (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
2120notbii 323 . 2 (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))
221, 13, 213bitr4g 317 1 (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2115  wrex 3127  cmpt 5119  ran crn 5529  suc csuc 6166   Fn wfn 6323  1-1wf1 6325  cfv 6328  (class class class)co 7130  ωcom 7555  2oc2o 8071   ·o comu 8075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-omul 8082
This theorem is referenced by:  fin1a2lem6  9804
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