![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s2f1o | Structured version Visualization version GIF version |
Description: A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.) |
Ref | Expression |
---|---|
s2f1o | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐸 = 〈“𝐴𝐵”〉 → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1084 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ∈ 𝑆) | |
2 | 0z 11426 | . . . . . 6 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | jctil 559 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ∈ ℤ ∧ 𝐴 ∈ 𝑆)) |
4 | simpl2 1085 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐵 ∈ 𝑆) | |
5 | 1z 11445 | . . . . . 6 ⊢ 1 ∈ ℤ | |
6 | 4, 5 | jctil 559 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆)) |
7 | 3, 6 | jca 553 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → ((0 ∈ ℤ ∧ 𝐴 ∈ 𝑆) ∧ (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆))) |
8 | simpl3 1086 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ≠ 𝐵) | |
9 | 0ne1 11126 | . . . . 5 ⊢ 0 ≠ 1 | |
10 | 8, 9 | jctil 559 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ≠ 1 ∧ 𝐴 ≠ 𝐵)) |
11 | f1oprg 6219 | . . . 4 ⊢ (((0 ∈ ℤ ∧ 𝐴 ∈ 𝑆) ∧ (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆)) → ((0 ≠ 1 ∧ 𝐴 ≠ 𝐵) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}–1-1-onto→{𝐴, 𝐵})) | |
12 | 7, 10, 11 | sylc 65 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}–1-1-onto→{𝐴, 𝐵}) |
13 | eqcom 2658 | . . . . . 6 ⊢ (𝐸 = 〈“𝐴𝐵”〉 ↔ 〈“𝐴𝐵”〉 = 𝐸) | |
14 | s2prop 13698 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) | |
15 | 14 | 3adant3 1101 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
16 | 15 | eqeq1d 2653 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (〈“𝐴𝐵”〉 = 𝐸 ↔ {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸)) |
17 | 13, 16 | syl5bb 272 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐸 = 〈“𝐴𝐵”〉 ↔ {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸)) |
18 | 17 | biimpa 500 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸) |
19 | eqidd 2652 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {0, 1} = {0, 1}) | |
20 | eqidd 2652 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {𝐴, 𝐵} = {𝐴, 𝐵}) | |
21 | 18, 19, 20 | f1oeq123d 6171 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → ({〈0, 𝐴〉, 〈1, 𝐵〉}:{0, 1}–1-1-onto→{𝐴, 𝐵} ↔ 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵})) |
22 | 12, 21 | mpbid 222 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵}) |
23 | 22 | ex 449 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐸 = 〈“𝐴𝐵”〉 → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 {cpr 4212 〈cop 4216 –1-1-onto→wf1o 5925 0cc0 9974 1c1 9975 ℤcz 11415 〈“cs2 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-concat 13333 df-s1 13334 df-s2 13639 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |