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Mirrors > Home > MPE Home > Th. List > fin1a2lem4 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9237. (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 7720 | . . . 4 ⊢ 2𝑜 ∈ ω | |
3 | nnmcl 7692 | . . . 4 ⊢ ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω) | |
4 | 2, 3 | mpan 706 | . . 3 ⊢ (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω) |
5 | 1, 4 | fmpti 6383 | . 2 ⊢ 𝐸:ω⟶ω |
6 | 1 | fin1a2lem3 9224 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2𝑜 ·𝑜 𝑎)) |
7 | 1 | fin1a2lem3 9224 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2𝑜 ·𝑜 𝑏)) |
8 | 6, 7 | eqeqan12d 2638 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏))) |
9 | 2on 7568 | . . . . . . 7 ⊢ 2𝑜 ∈ On | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On) |
11 | nnon 7071 | . . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On) | |
12 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) |
13 | nnon 7071 | . . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On) | |
14 | 13 | adantl 482 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On) |
15 | 0lt1o 7584 | . . . . . . . . 9 ⊢ ∅ ∈ 1𝑜 | |
16 | elelsuc 5797 | . . . . . . . . 9 ⊢ (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ ∅ ∈ suc 1𝑜 |
18 | df-2o 7561 | . . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜 | |
19 | 17, 18 | eleqtrri 2700 | . . . . . . 7 ⊢ ∅ ∈ 2𝑜 |
20 | 19 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜) |
21 | omcan 7649 | . . . . . 6 ⊢ (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏)) | |
22 | 10, 12, 14, 20, 21 | syl31anc 1329 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏)) |
23 | 8, 22 | bitrd 268 | . . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏)) |
24 | 23 | biimpd 219 | . . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)) |
25 | 24 | rgen2a 2977 | . 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏) |
26 | dff13 6512 | . 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))) | |
27 | 5, 25, 26 | mpbir2an 955 | 1 ⊢ 𝐸:ω–1-1→ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∅c0 3915 ↦ cmpt 4729 Oncon0 5723 suc csuc 5725 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 ωcom 7065 1𝑜c1o 7553 2𝑜c2o 7554 ·𝑜 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: fin1a2lem5 9226 fin1a2lem6 9227 fin1a2lem7 9228 |
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