Theorem fin1a2lem4 9511

Description: Lemma for fin1a2 9523. (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.

Step	Hyp	Ref	Expression
1		fin1a2lem.b	. . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
2		2onn 7958	. . . 4 ⊢ 2𝑜 ∈ ω
3		nnmcl 7930	. . . 4 ⊢ ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω)
4	2, 3	mpan 682	. . 3 ⊢ (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω)
5	1, 4	fmpti 6606	. 2 ⊢ 𝐸:ω⟶ω
6	1	fin1a2lem3 9510	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2𝑜 ·𝑜 𝑎))
7	1	fin1a2lem3 9510	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2𝑜 ·𝑜 𝑏))
8	6, 7	eqeqan12d 2813	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏)))
9		2on 7806	. . . . . . 7 ⊢ 2𝑜 ∈ On
10	9	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On)
11		nnon 7303	. . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
12	11	adantr 473	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13		nnon 7303	. . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
14	13	adantl 474	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15		0lt1o 7822	. . . . . . . . 9 ⊢ ∅ ∈ 1𝑜
16		elelsuc 6011	. . . . . . . . 9 ⊢ (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ ∅ ∈ suc 1𝑜
18		df-2o 7798	. . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜
19	17, 18	eleqtrri 2875	. . . . . . 7 ⊢ ∅ ∈ 2𝑜
20	19	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜)
21		omcan 7887	. . . . . 6 ⊢ (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
22	10, 12, 14, 20, 21	syl31anc 1493	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏))
23	8, 22	bitrd 271	. . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏))
24	23	biimpd 221	. . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))
25	24	rgen2a 3156	. 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)
26		dff13 6738	. 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)))
27	5, 25, 26	mpbir2an 703	1 ⊢ 𝐸:ω–1-1→ω

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3087  ∅c0 4113   ↦ cmpt 4920  Oncon0 5939  suc csuc 5941  ⟶wf 6095  –1-1→wf1 6096  ‘cfv 6099  (class class class)co 6876  ωcom 7297  1_𝑜c1o 7790  2_𝑜c2o 7791   ·_𝑜 comu 7795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-omul 7802

This theorem is referenced by:  fin1a2lem5  9512  fin1a2lem6  9513  fin1a2lem7  9514