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| Description: The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.) | 
| Ref | Expression | 
|---|---|
| s1rn | ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = {𝐴}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | s1val 14637 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
| 2 | 1 | rneqd 5948 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = ran {〈0, 𝐴〉}) | 
| 3 | c0ex 11256 | . . 3 ⊢ 0 ∈ V | |
| 4 | 3 | rnsnop 6243 | . 2 ⊢ ran {〈0, 𝐴〉} = {𝐴} | 
| 5 | 2, 4 | eqtrdi 2792 | 1 ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = {𝐴}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {csn 4625 〈cop 4631 ran crn 5685 0cc0 11156 〈“cs1 14634 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-mulcl 11218 ax-i2m1 11224 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fv 6568 df-s1 14635 | 
| This theorem is referenced by: s2rn 15003 s3rn 15004 s7rn 15005 cycpmco2f1 33145 cycpmco2rn 33146 unitprodclb 33418 mrsubvrs 35528 | 
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