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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 1187 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ∈ 𝑆) | |
2 | 0z 11995 | . . . . . 6 ⊢ 0 ∈ ℤ | |
3 | 1, 2 | jctil 522 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ∈ ℤ ∧ 𝐴 ∈ 𝑆)) |
4 | simpl2 1188 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐵 ∈ 𝑆) | |
5 | 1z 12015 | . . . . . 6 ⊢ 1 ∈ ℤ | |
6 | 4, 5 | jctil 522 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆)) |
7 | 3, 6 | jca 514 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → ((0 ∈ ℤ ∧ 𝐴 ∈ 𝑆) ∧ (1 ∈ ℤ ∧ 𝐵 ∈ 𝑆))) |
8 | simpl3 1189 | . . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐴 ≠ 𝐵) | |
9 | 0ne1 11711 | . . . . 5 ⊢ 0 ≠ 1 | |
10 | 8, 9 | jctil 522 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → (0 ≠ 1 ∧ 𝐴 ≠ 𝐵)) |
11 | f1oprg 6661 | . . . 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 2830 | . . . . . 6 ⊢ (𝐸 = 〈“𝐴𝐵”〉 ↔ 〈“𝐴𝐵”〉 = 𝐸) | |
14 | s2prop 14271 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) | |
15 | 14 | 3adant3 1128 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 〈“𝐴𝐵”〉 = {〈0, 𝐴〉, 〈1, 𝐵〉}) |
16 | 15 | eqeq1d 2825 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (〈“𝐴𝐵”〉 = 𝐸 ↔ {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸)) |
17 | 13, 16 | syl5bb 285 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐸 = 〈“𝐴𝐵”〉 ↔ {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸)) |
18 | 17 | biimpa 479 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {〈0, 𝐴〉, 〈1, 𝐵〉} = 𝐸) |
19 | eqidd 2824 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {0, 1} = {0, 1}) | |
20 | eqidd 2824 | . . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → {𝐴, 𝐵} = {𝐴, 𝐵}) | |
21 | 18, 19, 20 | f1oeq123d 6612 | . . 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 234 | . 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ 𝐸 = 〈“𝐴𝐵”〉) → 𝐸:{0, 1}–1-1-onto→{𝐴, 𝐵}) |
23 | 22 | ex 415 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {cpr 4571 〈cop 4575 –1-1-onto→wf1o 6356 0cc0 10539 1c1 10540 ℤcz 11984 〈“cs2 14205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 |
This theorem is referenced by: (None) |
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