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Mirrors > Home > MPE Home > Th. List > fin1a2lem4 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9837. (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 8266 | . . . 4 ⊢ 2o ∈ ω | |
3 | nnmcl 8238 | . . . 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 6876 | . 2 ⊢ 𝐸:ω⟶ω |
6 | 1 | fin1a2lem3 9824 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎)) |
7 | 1 | fin1a2lem3 9824 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2o ·o 𝑏)) |
8 | 6, 7 | eqeqan12d 2838 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏))) |
9 | 2on 8111 | . . . . . . 7 ⊢ 2o ∈ On | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On) |
11 | nnon 7586 | . . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On) | |
12 | 11 | adantr 483 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) |
13 | nnon 7586 | . . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On) | |
14 | 13 | adantl 484 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On) |
15 | 0lt1o 8129 | . . . . . . . . 9 ⊢ ∅ ∈ 1o | |
16 | elelsuc 6263 | . . . . . . . . 9 ⊢ (∅ ∈ 1o → ∅ ∈ suc 1o) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ ∅ ∈ suc 1o |
18 | df-2o 8103 | . . . . . . . 8 ⊢ 2o = suc 1o | |
19 | 17, 18 | eleqtrri 2912 | . . . . . . 7 ⊢ ∅ ∈ 2o |
20 | 19 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o) |
21 | omcan 8195 | . . . . . 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 3203 | . 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏) |
26 | dff13 7013 | . 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 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 ↦ cmpt 5146 Oncon0 6191 suc csuc 6193 ⟶wf 6351 –1-1→wf1 6352 ‘cfv 6355 (class class class)co 7156 ωcom 7580 1oc1o 8095 2oc2o 8096 ·o comu 8100 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 |
This theorem is referenced by: fin1a2lem5 9826 fin1a2lem6 9827 fin1a2lem7 9828 |
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