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Mirrors > Home > MPE Home > Th. List > fin1a2lem4 | Structured version Visualization version GIF version |
Description: Lemma for fin1a2 9120. (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 7607 | . . . 4 ⊢ 2𝑜 ∈ ω | |
3 | nnmcl 7579 | . . . 4 ⊢ ((2𝑜 ∈ ω ∧ 𝑥 ∈ ω) → (2𝑜 ·𝑜 𝑥) ∈ ω) | |
4 | 2, 3 | mpan 702 | . . 3 ⊢ (𝑥 ∈ ω → (2𝑜 ·𝑜 𝑥) ∈ ω) |
5 | 1, 4 | fmpti 6291 | . 2 ⊢ 𝐸:ω⟶ω |
6 | 1 | fin1a2lem3 9107 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2𝑜 ·𝑜 𝑎)) |
7 | 1 | fin1a2lem3 9107 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝐸‘𝑏) = (2𝑜 ·𝑜 𝑏)) |
8 | 6, 7 | eqeqan12d 2626 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ (2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏))) |
9 | 2on 7455 | . . . . . . 7 ⊢ 2𝑜 ∈ On | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 2𝑜 ∈ On) |
11 | nnon 6963 | . . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On) | |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑎 ∈ On) |
13 | nnon 6963 | . . . . . . 7 ⊢ (𝑏 ∈ ω → 𝑏 ∈ On) | |
14 | 13 | adantl 481 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → 𝑏 ∈ On) |
15 | 0lt1o 7471 | . . . . . . . . 9 ⊢ ∅ ∈ 1𝑜 | |
16 | elelsuc 5714 | . . . . . . . . 9 ⊢ (∅ ∈ 1𝑜 → ∅ ∈ suc 1𝑜) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . 8 ⊢ ∅ ∈ suc 1𝑜 |
18 | df-2o 7448 | . . . . . . . 8 ⊢ 2𝑜 = suc 1𝑜 | |
19 | 17, 18 | eleqtrri 2687 | . . . . . . 7 ⊢ ∅ ∈ 2𝑜 |
20 | 19 | a1i 11 | . . . . . 6 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ∅ ∈ 2𝑜) |
21 | omcan 7536 | . . . . . 6 ⊢ (((2𝑜 ∈ On ∧ 𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∅ ∈ 2𝑜) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏)) | |
22 | 10, 12, 14, 20, 21 | syl31anc 1321 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((2𝑜 ·𝑜 𝑎) = (2𝑜 ·𝑜 𝑏) ↔ 𝑎 = 𝑏)) |
23 | 8, 22 | bitrd 267 | . . . 4 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) ↔ 𝑎 = 𝑏)) |
24 | 23 | biimpd 218 | . . 3 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏)) |
25 | 24 | rgen2a 2960 | . 2 ⊢ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏) |
26 | dff13 6416 | . 2 ⊢ (𝐸:ω–1-1→ω ↔ (𝐸:ω⟶ω ∧ ∀𝑎 ∈ ω ∀𝑏 ∈ ω ((𝐸‘𝑎) = (𝐸‘𝑏) → 𝑎 = 𝑏))) | |
27 | 5, 25, 26 | mpbir2an 957 | 1 ⊢ 𝐸:ω–1-1→ω |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 ↦ cmpt 4643 Oncon0 5640 suc csuc 5642 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 ωcom 6957 1𝑜c1o 7440 2𝑜c2o 7441 ·𝑜 comu 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 |
This theorem is referenced by: fin1a2lem5 9109 fin1a2lem6 9110 fin1a2lem7 9111 |
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