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Theorem fin1a2lem4 9177
Description: Lemma for fin1a2 9189. (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 7672 . . . 4 2𝑜 ∈ ω
3 nnmcl 7644 . . . 4 ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω)
42, 3mpan 705 . . 3 (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω)
51, 4fmpti 6344 . 2 𝐸:ω⟶ω
61fin1a2lem3 9176 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2𝑜 ·𝑜 𝑎))
71fin1a2lem3 9176 . . . . . 6 (𝑏 ∈ ω → (𝐸𝑏) = (2𝑜 ·𝑜 𝑏))
86, 7eqeqan12d 2637 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏)))
9 2on 7520 . . . . . . 7 2𝑜 ∈ On
109a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On)
11 nnon 7025 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ On)
1211adantr 481 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13 nnon 7025 . . . . . . 7 (𝑏 ∈ ω → 𝑏 ∈ On)
1413adantl 482 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15 0lt1o 7536 . . . . . . . . 9 ∅ ∈ 1𝑜
16 elelsuc 5761 . . . . . . . . 9 (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜)
1715, 16ax-mp 5 . . . . . . . 8 ∅ ∈ suc 1𝑜
18 df-2o 7513 . . . . . . . 8 2𝑜 = suc 1𝑜
1917, 18eleqtrri 2697 . . . . . . 7 ∅ ∈ 2𝑜
2019a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜)
21 omcan 7601 . . . . . 6 (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
2210, 12, 14, 20, 21syl31anc 1326 . . . . 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 2972 . 2 𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)
26 dff13 6472 . 2 (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)))
275, 25, 26mpbir2an 954 1 𝐸:ω–1-1→ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  c0 3896  cmpt 4678  Oncon0 5687  suc csuc 5689  wf 5848  1-1wf1 5849  cfv 5852  (class class class)co 6610  ωcom 7019  1𝑜c1o 7505  2𝑜c2o 7506   ·𝑜 comu 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-omul 7517
This theorem is referenced by:  fin1a2lem5  9178  fin1a2lem6  9179  fin1a2lem7  9180
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