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Mirrors > Home > MPE Home > Th. List > rnsnop | Structured version Visualization version GIF version |
Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
cnvsn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rnsnop | ⊢ ran {〈𝐴, 𝐵〉} = {𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | rnsnopg 6045 | . 2 ⊢ (𝐴 ∈ V → ran {〈𝐴, 𝐵〉} = {𝐵}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran {〈𝐴, 𝐵〉} = {𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 ran crn 5520 |
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-v 3443 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-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: op2nda 6052 fpr 6893 en1 8559 fodomfi 8781 dcomex 9858 s1rn 13944 axlowdimlem13 26748 ex-rn 28225 ex-ima 28227 ptrest 35056 poimirlem3 35060 gidsn 35390 zrdivrng 35391 |
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