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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 14542 | . 2 ⊢ ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩} | |
2 | fvi 6964 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4879 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, 𝐴⟩) |
4 | 3 | sneqd 4639 | . 2 ⊢ (𝐴 ∈ 𝑉 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, 𝐴⟩}) |
5 | 1, 4 | eqtrid 2784 | 1 ⊢ (𝐴 ∈ 𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {csn 4627 ⟨cop 4633 I cid 5572 ‘cfv 6540 0cc0 11106 ⟨“cs1 14541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-s1 14542 |
This theorem is referenced by: s1rn 14545 s1cl 14548 s1dmALT 14555 s1fv 14556 s111 14561 repsw1 14729 s1co 14780 s2prop 14854 ofs1 14913 gsumws1 18715 uspgr1ewop 28494 usgr2v1e2w 28498 0wlkons1 29363 s1f1 32096 cshw1s2 32111 ofcs1 33543 signstf0 33567 |
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