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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 1188 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ∈ 𝑆) | |
2 | 0z 11980 | . . . . . 6 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | jctil 523 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ∈ ℤ ∧ 𝐴 ∈ 𝑆)) |
4 | simpl2 1189 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐵 ∈ 𝑆) | |
5 | 1z 12000 | . . . . . 6 ⊢ 1 ∈ ℤ | |
6 | 4, 5 | jctil 523 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆)) |
7 | 3, 6 | jca 515 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → ((0 ∈ ℤ ∧ 𝐴 ∈ 𝑆) ∧ (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆))) |
8 | simpl3 1190 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ≠ 𝐵) | |
9 | 0ne1 11696 | . . . . 5 ⊢ 0 ≠ 1 | |
10 | 8, 9 | jctil 523 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ≠ 1 ∧ 𝐴 ≠ 𝐵)) |
11 | f1oprg 6634 | . . . 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 2805 | . . . . . 6 ⊢ (𝐸 = 〈“𝐴𝐵”〉 ↔ 〈“𝐴𝐵”〉 = 𝐸) | |
14 | s2prop 14260 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) | |
15 | 14 | 3adant3 1129 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
16 | 15 | eqeq1d 2800 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (〈“𝐴𝐵”〉 = 𝐸 ↔ {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸)) |
17 | 13, 16 | syl5bb 286 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐸 = 〈“𝐴𝐵”〉 ↔ {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸)) |
18 | 17 | biimpa 480 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸) |
19 | eqidd 2799 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {0, 1} = {0, 1}) | |
20 | eqidd 2799 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {𝐴, 𝐵} = {𝐴, 𝐵}) | |
21 | 18, 19, 20 | f1oeq123d 6585 | . . 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 235 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵}) |
23 | 22 | ex 416 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {cpr 4527 〈cop 4531 –1-1-onto→wf1o 6323 0cc0 10526 1c1 10527 ℤcz 11969 〈“cs2 14194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 |
This theorem is referenced by: (None) |
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