Theorem fin1a2lem4 10440

Description: Lemma for fin1a2 10452. (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 8678	. . . 4 ⊢ 2o ∈ ω
3		nnmcl 8648	. . . 4 ⊢ ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω)
4	2, 3	mpan 690	. . 3 ⊢ (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω)
5	1, 4	fmpti 7131	. 2 ⊢ 𝐸:ω⟶ω
6	1	fin1a2lem3 10439	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎))
7	1	fin1a2lem3 10439	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2o ·o 𝑏))
8	6, 7	eqeqan12d 2748	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9		2on 8518	. . . . . . 7 ⊢ 2o ∈ On
10	9	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11		nnon 7892	. . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
12	11	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13		nnon 7892	. . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
14	13	adantl 481	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15		0lt1o 8540	. . . . . . . . 9 ⊢ ∅ ∈ 1o
16		elelsuc 6458	. . . . . . . . 9 ⊢ (∅ ∈ 1o → ∅ ∈ suc 1o)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ ∅ ∈ suc 1o
18		df-2o 8505	. . . . . . . 8 ⊢ 2o = suc 1o
19	17, 18	eleqtrri 2837	. . . . . . 7 ⊢ ∅ ∈ 2o
20	19	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21		omcan 8605	. . . . . 6 ⊢ (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
22	10, 12, 14, 20, 21	syl31anc 1372	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
23	8, 22	bitrd 279	. . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏))
24	23	biimpd 229	. . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))
25	24	rgen2 3196	. 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)
26		dff13 7274	. 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)))
27	5, 25, 26	mpbir2an 711	1 ⊢ 𝐸:ω–1-1→ω

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∅c0 4338   ↦ cmpt 5230  Oncon0 6385  suc csuc 6387  ⟶wf 6558  –1-1→wf1 6559  ‘cfv 6562  (class class class)co 7430  ωcom 7886  1_oc1o 8497  2_oc2o 8498   ·_o comu 8502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509

This theorem is referenced by:  fin1a2lem5  10441  fin1a2lem6  10442  fin1a2lem7  10443