Theorem fin1a2lem4 10289

Description: Lemma for fin1a2 10301. (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 8552	. . . 4 ⊢ 2o ∈ ω
3		nnmcl 8522	. . . 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 7040	. 2 ⊢ 𝐸:ω⟶ω
6	1	fin1a2lem3 10288	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎))
7	1	fin1a2lem3 10288	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2o ·o 𝑏))
8	6, 7	eqeqan12d 2745	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9		2on 8393	. . . . . . 7 ⊢ 2o ∈ On
10	9	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11		nnon 7797	. . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
12	11	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13		nnon 7797	. . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
14	13	adantl 481	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15		0lt1o 8414	. . . . . . . . 9 ⊢ ∅ ∈ 1o
16		elelsuc 6376	. . . . . . . . 9 ⊢ (∅ ∈ 1o → ∅ ∈ suc 1o)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ ∅ ∈ suc 1o
18		df-2o 8381	. . . . . . . 8 ⊢ 2o = suc 1o
19	17, 18	eleqtrri 2830	. . . . . . 7 ⊢ ∅ ∈ 2o
20	19	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21		omcan 8479	. . . . . 6 ⊢ (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
22	10, 12, 14, 20, 21	syl31anc 1375	. . . . 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 3172	. 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)
26		dff13 7183	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∅c0 4278   ↦ cmpt 5167  Oncon0 6301  suc csuc 6303  ⟶wf 6472  –1-1→wf1 6473  ‘cfv 6476  (class class class)co 7341  ωcom 7791  1_oc1o 8373  2_oc2o 8374   ·_o comu 8378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385

This theorem is referenced by:  fin1a2lem5  10290  fin1a2lem6  10291  fin1a2lem7  10292