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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 14631 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | fvi 6985 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4885 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈0, ( I ‘𝐴)〉 = 〈0, 𝐴〉) |
4 | 3 | sneqd 4643 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈0, ( I ‘𝐴)〉} = {〈0, 𝐴〉}) |
5 | 1, 4 | eqtrid 2787 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 I cid 5582 ‘cfv 6563 0cc0 11153 〈“cs1 14630 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-s1 14631 |
This theorem is referenced by: s1rn 14634 s1cl 14637 s1dmALT 14644 s1fv 14645 s111 14650 repsw1 14818 s1co 14869 s2prop 14943 ofs1 15006 gsumws1 18864 uspgr1ewop 29280 usgr2v1e2w 29284 0wlkons1 30150 s1f1 32912 cshw1s2 32930 ofcs1 34538 signstf0 34562 |
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