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Theorem fin1a2lem4 9824
Description: Lemma for fin1a2 9836. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.b 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
Assertion
Ref Expression
fin1a2lem4 𝐸:ω–1-1→ω

Proof of Theorem fin1a2lem4
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
2 2onn 8265 . . . 4 2o ∈ ω
3 nnmcl 8237 . . . 4 ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω)
42, 3mpan 688 . . 3 (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω)
51, 4fmpti 6875 . 2 𝐸:ω⟶ω
61fin1a2lem3 9823 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2o ·o 𝑎))
71fin1a2lem3 9823 . . . . . 6 (𝑏 ∈ ω → (𝐸𝑏) = (2o ·o 𝑏))
86, 7eqeqan12d 2838 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9 2on 8110 . . . . . . 7 2o ∈ On
109a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11 nnon 7585 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ On)
1211adantr 483 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13 nnon 7585 . . . . . . 7 (𝑏 ∈ ω → 𝑏 ∈ On)
1413adantl 484 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15 0lt1o 8128 . . . . . . . . 9 ∅ ∈ 1o
16 elelsuc 6262 . . . . . . . . 9 (∅ ∈ 1o → ∅ ∈ suc 1o)
1715, 16ax-mp 5 . . . . . . . 8 ∅ ∈ suc 1o
18 df-2o 8102 . . . . . . . 8 2o = suc 1o
1917, 18eleqtrri 2912 . . . . . . 7 ∅ ∈ 2o
2019a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21 omcan 8194 . . . . . 6 (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
2210, 12, 14, 20, 21syl31anc 1369 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
238, 22bitrd 281 . . . 4 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ 𝑎 = 𝑏))
2423biimpd 231 . . 3 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏))
2524rgen2 3203 . 2 𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)
26 dff13 7012 . 2 (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)))
275, 25, 26mpbir2an 709 1 𝐸:ω–1-1→ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  c0 4290  cmpt 5145  Oncon0 6190  suc csuc 6192  wf 6350  1-1wf1 6351  cfv 6354  (class class class)co 7155  ωcom 7579  1oc1o 8094  2oc2o 8095   ·o comu 8099
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-omul 8106
This theorem is referenced by:  fin1a2lem5  9825  fin1a2lem6  9826  fin1a2lem7  9827
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