Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6072 | . 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 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 〈cop 4566 ran crn 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 |
This theorem is referenced by: op2nda 6079 fpr 6910 en1 8570 fodomfi 8791 dcomex 9863 s1rn 13947 axlowdimlem13 26734 ex-rn 28213 ex-ima 28215 ptrest 34885 poimirlem3 34889 gidsn 35224 zrdivrng 35225 |
Copyright terms: Public domain | W3C validator |