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Theorem fin1a2lem4 10443
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
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 8680 . . . 4 2o ∈ ω
3 nnmcl 8650 . . . 4 ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω)
42, 3mpan 690 . . 3 (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω)
51, 4fmpti 7132 . 2 𝐸:ω⟶ω
61fin1a2lem3 10442 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2o ·o 𝑎))
71fin1a2lem3 10442 . . . . . 6 (𝑏 ∈ ω → (𝐸𝑏) = (2o ·o 𝑏))
86, 7eqeqan12d 2751 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9 2on 8520 . . . . . . 7 2o ∈ On
109a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11 nnon 7893 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ On)
1211adantr 480 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13 nnon 7893 . . . . . . 7 (𝑏 ∈ ω → 𝑏 ∈ On)
1413adantl 481 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15 0lt1o 8542 . . . . . . . . 9 ∅ ∈ 1o
16 elelsuc 6457 . . . . . . . . 9 (∅ ∈ 1o → ∅ ∈ suc 1o)
1715, 16ax-mp 5 . . . . . . . 8 ∅ ∈ suc 1o
18 df-2o 8507 . . . . . . . 8 2o = suc 1o
1917, 18eleqtrri 2840 . . . . . . 7 ∅ ∈ 2o
2019a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21 omcan 8607 . . . . . 6 (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
2210, 12, 14, 20, 21syl31anc 1375 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
238, 22bitrd 279 . . . 4 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ 𝑎 = 𝑏))
2423biimpd 229 . . 3 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏))
2524rgen2 3199 . 2 𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)
26 dff13 7275 . 2 (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)))
275, 25, 26mpbir2an 711 1 𝐸:ω–1-1→ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  c0 4333  cmpt 5225  Oncon0 6384  suc csuc 6386  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  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:  fin1a2lem5  10444  fin1a2lem6  10445  fin1a2lem7  10446
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