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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 13950 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | fvi 6740 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | opeq2d 4810 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈0, ( I ‘𝐴)〉 = 〈0, 𝐴〉) |
4 | 3 | sneqd 4579 | . 2 ⊢ (𝐴 ∈ 𝑉 → {〈0, ( I ‘𝐴)〉} = {〈0, 𝐴〉}) |
5 | 1, 4 | syl5eq 2868 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4567 〈cop 4573 I cid 5459 ‘cfv 6355 0cc0 10537 〈“cs1 13949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-s1 13950 |
This theorem is referenced by: s1rn 13953 s1cl 13956 s1dmALT 13963 s1fv 13964 s111 13969 repsw1 14145 s1co 14195 s2prop 14269 ofs1 14330 gsumws1 18002 uspgr1ewop 27030 usgr2v1e2w 27034 0wlkons1 27900 s1f1 30619 cshw1s2 30634 ofcs1 31814 signstf0 31838 |
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