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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 13941 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | fvi 6715 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4772 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈0, ( I ‘𝐴)〉 = 〈0, 𝐴〉) |
4 | 3 | sneqd 4537 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈0, ( I ‘𝐴)〉} = {〈0, 𝐴〉}) |
5 | 1, 4 | syl5eq 2845 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 I cid 5424 ‘cfv 6324 0cc0 10526 〈“cs1 13940 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-s1 13941 |
This theorem is referenced by: s1rn 13944 s1cl 13947 s1dmALT 13954 s1fv 13955 s111 13960 repsw1 14136 s1co 14186 s2prop 14260 ofs1 14321 gsumws1 17994 uspgr1ewop 27038 usgr2v1e2w 27042 0wlkons1 27906 s1f1 30645 cshw1s2 30660 ofcs1 31924 signstf0 31948 |
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