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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 1064 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ∈ 𝑆) | |
2 | 0z 11388 | . . . . . 6 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | jctil 560 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ∈ ℤ ∧ 𝐴 ∈ 𝑆)) |
4 | simpl2 1065 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐵 ∈ 𝑆) | |
5 | 1z 11407 | . . . . . 6 ⊢ 1 ∈ ℤ | |
6 | 4, 5 | jctil 560 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆)) |
7 | 3, 6 | jca 554 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → ((0 ∈ ℤ ∧ 𝐴 ∈ 𝑆) ∧ (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆))) |
8 | simpl3 1066 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ≠ 𝐵) | |
9 | 0ne1 11088 | . . . . 5 ⊢ 0 ≠ 1 | |
10 | 8, 9 | jctil 560 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ≠ 1 ∧ 𝐴 ≠ 𝐵)) |
11 | f1oprg 6181 | . . . 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 2629 | . . . . . 6 ⊢ (𝐸 = 〈“𝐴𝐵”〉 ↔ 〈“𝐴𝐵”〉 = 𝐸) | |
14 | s2prop 13652 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) | |
15 | 14 | 3adant3 1081 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
16 | 15 | eqeq1d 2624 | . . . . . 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 501 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸) |
19 | eqidd 2623 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {0, 1} = {0, 1}) | |
20 | eqidd 2623 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {𝐴, 𝐵} = {𝐴, 𝐵}) | |
21 | 18, 19, 20 | f1oeq123d 6133 | . . 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 450 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {cpr 4179 〈cop 4183 –1-1-onto→wf1o 5887 0cc0 9936 1c1 9937 ℤcz 11377 〈“cs2 13586 |
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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-nel 2898 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-int 4476 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 |
This theorem is referenced by: (None) |
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