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Mirrors > Home > MPE Home > Th. List > s1val | Structured version Visualization version GIF version |
Description: Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1val | ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s1 13944 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | fvi 6734 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4803 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈0, ( I ‘𝐴)〉 = 〈0, 𝐴〉) |
4 | 3 | sneqd 4572 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈0, ( I ‘𝐴)〉} = {〈0, 𝐴〉}) |
5 | 1, 4 | syl5eq 2868 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4560 〈cop 4566 I cid 5453 ‘cfv 6349 0cc0 10531 〈“cs1 13943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-s1 13944 |
This theorem is referenced by: s1rn 13947 s1cl 13950 s1dmALT 13957 s1fv 13958 s111 13963 repsw1 14139 s1co 14189 s2prop 14263 ofs1 14324 gsumws1 17996 uspgr1ewop 27024 usgr2v1e2w 27028 0wlkons1 27894 s1f1 30614 cshw1s2 30629 ofcs1 31809 signstf0 31833 |
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