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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 14491 | . 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩} | |
2 | fvi 6922 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4842 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩) |
4 | 3 | sneqd 4603 | . 2 ⊢ (𝐴 ∈ 𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩}) |
5 | 1, 4 | eqtrid 2789 | 1 ⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4591 ⟨cop 4597 I cid 5535 ‘cfv 6501 0cc0 11058 ⟨“cs1 14490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-s1 14491 |
This theorem is referenced by: s1rn 14494 s1cl 14497 s1dmALT 14504 s1fv 14505 s111 14510 repsw1 14678 s1co 14729 s2prop 14803 ofs1 14862 gsumws1 18655 uspgr1ewop 28238 usgr2v1e2w 28242 0wlkons1 29107 s1f1 31841 cshw1s2 31856 ofcs1 33196 signstf0 33220 |
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