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Mirrors > Home > MPE Home > Th. List > s1rn | Structured version Visualization version GIF version |
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 14545 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | rneqd 5927 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = ran {〈0, 𝐴〉}) |
3 | c0ex 11205 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | rnsnop 6213 | . 2 ⊢ ran {〈0, 𝐴〉} = {𝐴} |
5 | 2, 4 | eqtrdi 2780 | 1 ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4620 〈cop 4626 ran crn 5667 0cc0 11106 〈“cs1 14542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-mulcl 11168 ax-i2m1 11174 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6485 df-fun 6535 df-fv 6541 df-s1 14543 |
This theorem is referenced by: cycpmco2f1 32751 cycpmco2rn 32752 mrsubvrs 35002 |
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