Theorem fin1a2lem4 9828

Description: Lemma for fin1a2 9840. (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 8269	. . . 4 ⊢ 2o ∈ ω
3		nnmcl 8241	. . . 4 ⊢ ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω)
4	2, 3	mpan 688	. . 3 ⊢ (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω)
5	1, 4	fmpti 6879	. 2 ⊢ 𝐸:ω⟶ω
6	1	fin1a2lem3 9827	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎))
7	1	fin1a2lem3 9827	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2o ·o 𝑏))
8	6, 7	eqeqan12d 2841	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9		2on 8114	. . . . . . 7 ⊢ 2o ∈ On
10	9	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11		nnon 7589	. . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
12	11	adantr 483	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13		nnon 7589	. . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On)
14	13	adantl 484	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15		0lt1o 8132	. . . . . . . . 9 ⊢ ∅ ∈ 1o
16		elelsuc 6266	. . . . . . . . 9 ⊢ (∅ ∈ 1o → ∅ ∈ suc 1o)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ ∅ ∈ suc 1o
18		df-2o 8106	. . . . . . . 8 ⊢ 2o = suc 1o
19	17, 18	eleqtrri 2915	. . . . . . 7 ⊢ ∅ ∈ 2o
20	19	a1i 11	. . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21		omcan 8198	. . . . . 6 ⊢ (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
22	10, 12, 14, 20, 21	syl31anc 1369	. . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
23	8, 22	bitrd 281	. . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏))
24	23	biimpd 231	. . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))
25	24	rgen2 3206	. 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)
26		dff13 7016	. 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)))
27	5, 25, 26	mpbir2an 709	1 ⊢ 𝐸:ω–1-1→ω

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∅c0 4294   ↦ cmpt 5149  Oncon0 6194  suc csuc 6196  ⟶wf 6354  –1-1→wf1 6355  ‘cfv 6358  (class class class)co 7159  ωcom 7583  1_oc1o 8098  2_oc2o 8099   ·_o comu 8103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-omul 8110

This theorem is referenced by:  fin1a2lem5  9829  fin1a2lem6  9830  fin1a2lem7  9831