![]() |
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 14582 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | fvi 6973 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4882 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈0, ( I ‘𝐴)〉 = 〈0, 𝐴〉) |
4 | 3 | sneqd 4642 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈0, ( I ‘𝐴)〉} = {〈0, 𝐴〉}) |
5 | 1, 4 | eqtrid 2777 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4630 〈cop 4636 I cid 5575 ‘cfv 6549 0cc0 11140 〈“cs1 14581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-s1 14582 |
This theorem is referenced by: s1rn 14585 s1cl 14588 s1dmALT 14595 s1fv 14596 s111 14601 repsw1 14769 s1co 14820 s2prop 14894 ofs1 14953 gsumws1 18798 uspgr1ewop 29133 usgr2v1e2w 29137 0wlkons1 30003 s1f1 32753 cshw1s2 32770 ofcs1 34307 signstf0 34331 |
Copyright terms: Public domain | W3C validator |