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Theorem fin1a2lem4 9853
 Description: Lemma for fin1a2 9865. (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.
StepHypRef Expression
1 fin1a2lem.b . . 3 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥))
2 2onn 8274 . . . 4 2o ∈ ω
3 nnmcl 8246 . . . 4 ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω)
42, 3mpan 690 . . 3 (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω)
51, 4fmpti 6865 . 2 𝐸:ω⟶ω
61fin1a2lem3 9852 . . . . . 6 (𝑎 ∈ ω → (𝐸𝑎) = (2o ·o 𝑎))
71fin1a2lem3 9852 . . . . . 6 (𝑏 ∈ ω → (𝐸𝑏) = (2o ·o 𝑏))
86, 7eqeqan12d 2776 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏)))
9 2on 8119 . . . . . . 7 2o ∈ On
109a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On)
11 nnon 7583 . . . . . . 7 (𝑎 ∈ ω → 𝑎 ∈ On)
1211adantr 485 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
13 nnon 7583 . . . . . . 7 (𝑏 ∈ ω → 𝑏 ∈ On)
1413adantl 486 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On)
15 0lt1o 8137 . . . . . . . . 9 ∅ ∈ 1o
16 elelsuc 6239 . . . . . . . . 9 (∅ ∈ 1o → ∅ ∈ suc 1o)
1715, 16ax-mp 5 . . . . . . . 8 ∅ ∈ suc 1o
18 df-2o 8111 . . . . . . . 8 2o = suc 1o
1917, 18eleqtrri 2852 . . . . . . 7 ∅ ∈ 2o
2019a1i 11 . . . . . 6 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o)
21 omcan 8203 . . . . . 6 (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
2210, 12, 14, 20, 21syl31anc 1371 . . . . 5 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏))
238, 22bitrd 282 . . . 4 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) ↔ 𝑎 = 𝑏))
2423biimpd 232 . . 3 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏))
2524rgen2 3133 . 2 𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)
26 dff13 7003 . 2 (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸𝑎) = (𝐸𝑏) → 𝑎 = 𝑏)))
275, 25, 26mpbir2an 711 1 𝐸:ω–1-1→ω
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∅c0 4226   ↦ cmpt 5110  Oncon0 6167  suc csuc 6169  ⟶wf 6329  –1-1→wf1 6330  ‘cfv 6333  (class class class)co 7148  ωcom 7577  1oc1o 8103  2oc2o 8104   ·o comu 8108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-1o 8110  df-2o 8111  df-oadd 8114  df-omul 8115 This theorem is referenced by:  fin1a2lem5  9854  fin1a2lem6  9855  fin1a2lem7  9856
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