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Theorem fin1a2lem4 9430
Description: Lemma for fin1a2 9442. (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 7877 . . . 4 2𝑜 ∈ ω
3 nnmcl 7849 . . . 4 ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω)
42, 3mpan 670 . . 3 (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω)
51, 4fmpti 6527 . 2 𝐸:ω⟶ω
61fin1a2lem3 9429 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2𝑜 ·𝑜 𝑎))
71fin1a2lem3 9429 . . . . . 6 (𝑏 ∈ ω → (𝐸𝑏) = (2𝑜 ·𝑜 𝑏))
86, 7eqeqan12d 2787 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏)))
9 2on 7725 . . . . . . 7 2𝑜 ∈ On
109a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On)
11 nnon 7221 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ On)
1211adantr 466 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13 nnon 7221 . . . . . . 7 (𝑏 ∈ ω → 𝑏 ∈ On)
1413adantl 467 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15 0lt1o 7741 . . . . . . . . 9 ∅ ∈ 1𝑜
16 elelsuc 5939 . . . . . . . . 9 (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜)
1715, 16ax-mp 5 . . . . . . . 8 ∅ ∈ suc 1𝑜
18 df-2o 7717 . . . . . . . 8 2𝑜 = suc 1𝑜
1917, 18eleqtrri 2849 . . . . . . 7 ∅ ∈ 2𝑜
2019a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜)
21 omcan 7806 . . . . . 6 (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
2210, 12, 14, 20, 21syl31anc 1479 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
238, 22bitrd 268 . . . 4 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ 𝑎 = 𝑏))
2423biimpd 219 . . 3 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏))
2524rgen2a 3126 . 2 𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)
26 dff13 6657 . 2 (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)))
275, 25, 26mpbir2an 690 1 𝐸:ω–1-1→ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  c0 4063  cmpt 4864  Oncon0 5865  suc csuc 5867  wf 6026  1-1wf1 6027  cfv 6030  (class class class)co 6795  ωcom 7215  1𝑜c1o 7709  2𝑜c2o 7710   ·𝑜 comu 7714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-2o 7717  df-oadd 7720  df-omul 7721
This theorem is referenced by:  fin1a2lem5  9431  fin1a2lem6  9432  fin1a2lem7  9433
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