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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 13940 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
2 | 1 | rneqd 5801 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = ran {〈0, 𝐴〉}) |
3 | c0ex 10623 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | rnsnop 6074 | . 2 ⊢ ran {〈0, 𝐴〉} = {𝐴} |
5 | 2, 4 | syl6eq 2869 | 1 ⊢ (𝐴 ∈ 𝑉 → ran 〈“𝐴”〉 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 ran crn 5549 0cc0 10525 〈“cs1 13937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-s1 13938 |
This theorem is referenced by: cycpmco2f1 30693 cycpmco2rn 30694 mrsubvrs 32666 |
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