Theorem fin1a2lem4 10316

Description: Lemma for fin1a2 10328. (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 8571	. . . 4 ⊢ 2o ∈ ω
3		nnmcl 8541	. . . 4 ⊢ ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω)
4	2, 3	mpan 691	. . 3 ⊢ (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω)
5	1, 4	fmpti 7058	. 2 ⊢ 𝐸:ω⟶ω
6	1	fin1a2lem3 10315	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎))
7	1	fin1a2lem3 10315	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2o ·o 𝑏))
8	6, 7	eqeqan12d 2751	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9		2on 8411	. . . . . . 7 ⊢ 2o ∈ On
10	9	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11		nnon 7816	. . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
12	11	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13		nnon 7816	. . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
14	13	adantl 481	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15		0lt1o 8432	. . . . . . . . 9 ⊢ ∅ ∈ 1o
16		elelsuc 6392	. . . . . . . . 9 ⊢ (∅ ∈ 1o → ∅ ∈ suc 1o)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ ∅ ∈ suc 1o
18		df-2o 8399	. . . . . . . 8 ⊢ 2o = suc 1o
19	17, 18	eleqtrri 2836	. . . . . . 7 ⊢ ∅ ∈ 2o
20	19	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21		omcan 8497	. . . . . 6 ⊢ (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
22	10, 12, 14, 20, 21	syl31anc 1376	. . . . 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 3178	. 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)
26		dff13 7202	. 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)))
27	5, 25, 26	mpbir2an 712	1 ⊢ 𝐸:ω–1-1→ω

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∅c0 4274   ↦ cmpt 5167  Oncon0 6317  suc csuc 6319  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7360  ωcom 7810  1_oc1o 8391  2_oc2o 8392   ·_o comu 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-omul 8403

This theorem is referenced by:  fin1a2lem5  10317  fin1a2lem6  10318  fin1a2lem7  10319