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Mirrors > Home > MPE Home > Th. List > fin1a2lem4 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9523. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Ref | Expression |
---|---|
fin1a2lem.b | ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥)) |
Ref | Expression |
---|---|
fin1a2lem4 | ⊢ 𝐸:ω–1-1→ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fin1a2lem.b | . . 3 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥)) | |
2 | 2onn 7958 | . . . 4 ⊢ 2𝑜 ∈ ω | |
3 | nnmcl 7930 | . . . 4 ⊢ ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω) | |
4 | 2, 3 | mpan 682 | . . 3 ⊢ (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω) |
5 | 1, 4 | fmpti 6606 | . 2 ⊢ 𝐸:ω⟶ω |
6 | 1 | fin1a2lem3 9510 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2𝑜 ·𝑜 𝑎)) |
7 | 1 | fin1a2lem3 9510 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2𝑜 ·𝑜 𝑏)) |
8 | 6, 7 | eqeqan12d 2813 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏))) |
9 | 2on 7806 | . . . . . . 7 ⊢ 2𝑜 ∈ On | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On) |
11 | nnon 7303 | . . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On) | |
12 | 11 | adantr 473 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) |
13 | nnon 7303 | . . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On) | |
14 | 13 | adantl 474 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On) |
15 | 0lt1o 7822 | . . . . . . . . 9 ⊢ ∅ ∈ 1𝑜 | |
16 | elelsuc 6011 | . . . . . . . . 9 ⊢ (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ ∅ ∈ suc 1𝑜 |
18 | df-2o 7798 | . . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜 | |
19 | 17, 18 | eleqtrri 2875 | . . . . . . 7 ⊢ ∅ ∈ 2𝑜 |
20 | 19 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜) |
21 | omcan 7887 | . . . . . 6 ⊢ (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏)) | |
22 | 10, 12, 14, 20, 21 | syl31anc 1493 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏)) |
23 | 8, 22 | bitrd 271 | . . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏)) |
24 | 23 | biimpd 221 | . . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)) |
25 | 24 | rgen2a 3156 | . 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏) |
26 | dff13 6738 | . 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))) | |
27 | 5, 25, 26 | mpbir2an 703 | 1 ⊢ 𝐸:ω–1-1→ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3087 ∅c0 4113 ↦ cmpt 4920 Oncon0 5939 suc csuc 5941 ⟶wf 6095 –1-1→wf1 6096 ‘cfv 6099 (class class class)co 6876 ωcom 7297 1𝑜c1o 7790 2𝑜c2o 7791 ·𝑜 comu 7795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-omul 7802 |
This theorem is referenced by: fin1a2lem5 9512 fin1a2lem6 9513 fin1a2lem7 9514 |
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