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| Description: Lemma for fin1a2 10455. (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 8680 | . . . 4 ⊢ 2o ∈ ω | |
| 3 | nnmcl 8650 | . . . 4 ⊢ ((2o ∈ ω ∧ 𝑥 ∈ ω) → (2o ·o 𝑥) ∈ ω) | |
| 4 | 2, 3 | mpan 690 | . . 3 ⊢ (𝑥 ∈ ω → (2o ·o 𝑥) ∈ ω) | 
| 5 | 1, 4 | fmpti 7132 | . 2 ⊢ 𝐸:ω⟶ω | 
| 6 | 1 | fin1a2lem3 10442 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎)) | 
| 7 | 1 | fin1a2lem3 10442 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2o ·o 𝑏)) | 
| 8 | 6, 7 | eqeqan12d 2751 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2o ·o 𝑎) = (2o ·o 𝑏))) | 
| 9 | 2on 8520 | . . . . . . 7 ⊢ 2o ∈ On | |
| 10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2o ∈ On) | 
| 11 | nnon 7893 | . . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On) | |
| 12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) | 
| 13 | nnon 7893 | . . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On) | |
| 14 | 13 | adantl 481 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On) | 
| 15 | 0lt1o 8542 | . . . . . . . . 9 ⊢ ∅ ∈ 1o | |
| 16 | elelsuc 6457 | . . . . . . . . 9 ⊢ (∅ ∈ 1o → ∅ ∈ suc 1o) | |
| 17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ ∅ ∈ suc 1o | 
| 18 | df-2o 8507 | . . . . . . . 8 ⊢ 2o = suc 1o | |
| 19 | 17, 18 | eleqtrri 2840 | . . . . . . 7 ⊢ ∅ ∈ 2o | 
| 20 | 19 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2o) | 
| 21 | omcan 8607 | . . . . . 6 ⊢ (((2o ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2o) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏)) | |
| 22 | 10, 12, 14, 20, 21 | syl31anc 1375 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2o ·o 𝑎) = (2o ·o 𝑏) ↔ 𝑎 = 𝑏)) | 
| 23 | 8, 22 | bitrd 279 | . . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏)) | 
| 24 | 23 | biimpd 229 | . . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)) | 
| 25 | 24 | rgen2 3199 | . 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏) | 
| 26 | dff13 7275 | . 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))) | |
| 27 | 5, 25, 26 | mpbir2an 711 | 1 ⊢ 𝐸:ω–1-1→ω | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 ↦ cmpt 5225 Oncon0 6384 suc csuc 6386 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 ωcom 7887 1oc1o 8499 2oc2o 8500 ·o comu 8504 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 | 
| This theorem is referenced by: fin1a2lem5 10444 fin1a2lem6 10445 fin1a2lem7 10446 | 
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