![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14546 | . 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩} | |
2 | fvi 6968 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4881 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩) |
4 | 3 | sneqd 4641 | . 2 ⊢ (𝐴 ∈ 𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩}) |
5 | 1, 4 | eqtrid 2785 | 1 ⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4629 ⟨cop 4635 I cid 5574 ‘cfv 6544 0cc0 11110 ⟨“cs1 14545 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-s1 14546 |
This theorem is referenced by: s1rn 14549 s1cl 14552 s1dmALT 14559 s1fv 14560 s111 14565 repsw1 14733 s1co 14784 s2prop 14858 ofs1 14917 gsumws1 18719 uspgr1ewop 28505 usgr2v1e2w 28509 0wlkons1 29374 s1f1 32109 cshw1s2 32124 ofcs1 33555 signstf0 33579 |
Copyright terms: Public domain | W3C validator |