Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > s1f1 | Structured version Visualization version GIF version |
Description: Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.) |
Ref | Expression |
---|---|
s1f1.1 | ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
Ref | Expression |
---|---|
s1f1 | ⊢ (𝜑 → 〈“𝐼”〉:dom 〈“𝐼”〉–1-1→𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11910 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℕ0) |
3 | s1f1.1 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐷) | |
4 | f1osng 6652 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 𝐼 ∈ 𝐷) → {〈0, 𝐼〉}:{0}–1-1-onto→{𝐼}) | |
5 | 2, 3, 4 | syl2anc 586 | . . . 4 ⊢ (𝜑 → {〈0, 𝐼〉}:{0}–1-1-onto→{𝐼}) |
6 | f1of1 6611 | . . . 4 ⊢ ({〈0, 𝐼〉}:{0}–1-1-onto→{𝐼} → {〈0, 𝐼〉}:{0}–1-1→{𝐼}) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → {〈0, 𝐼〉}:{0}–1-1→{𝐼}) |
8 | 3 | snssd 4739 | . . 3 ⊢ (𝜑 → {𝐼} ⊆ 𝐷) |
9 | f1ss 6577 | . . 3 ⊢ (({〈0, 𝐼〉}:{0}–1-1→{𝐼} ∧ {𝐼} ⊆ 𝐷) → {〈0, 𝐼〉}:{0}–1-1→𝐷) | |
10 | 7, 8, 9 | syl2anc 586 | . 2 ⊢ (𝜑 → {〈0, 𝐼〉}:{0}–1-1→𝐷) |
11 | s1val 13948 | . . . 4 ⊢ (𝐼 ∈ 𝐷 → 〈“𝐼”〉 = {〈0, 𝐼〉}) | |
12 | 3, 11 | syl 17 | . . 3 ⊢ (𝜑 → 〈“𝐼”〉 = {〈0, 𝐼〉}) |
13 | s1dm 13958 | . . . 4 ⊢ dom 〈“𝐼”〉 = {0} | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → dom 〈“𝐼”〉 = {0}) |
15 | eqidd 2821 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐷) | |
16 | 12, 14, 15 | f1eq123d 6605 | . 2 ⊢ (𝜑 → (〈“𝐼”〉:dom 〈“𝐼”〉–1-1→𝐷 ↔ {〈0, 𝐼〉}:{0}–1-1→𝐷)) |
17 | 10, 16 | mpbird 259 | 1 ⊢ (𝜑 → 〈“𝐼”〉:dom 〈“𝐼”〉–1-1→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3933 {csn 4564 〈cop 4570 dom cdm 5552 –1-1→wf1 6349 –1-1-onto→wf1o 6351 0cc0 10534 ℕ0cn0 11895 〈“cs1 13945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-n0 11896 df-z 11980 df-uz 12242 df-fz 12891 df-fzo 13032 df-hash 13689 df-word 13860 df-s1 13946 |
This theorem is referenced by: cycpmco2f1 30787 |
Copyright terms: Public domain | W3C validator |