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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 6220 | . 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 1541 ∈ wcel 2106 Vcvv 3474 {csn 4628 ⟨cop 4634 ran crn 5677 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 |
This theorem is referenced by: op2nda 6227 fpr 7151 en1 9020 en1OLD 9021 fodomfi 9324 dcomex 10441 s1rn 14548 axlowdimlem13 28209 ex-rn 29690 ex-ima 29692 ptrest 36482 poimirlem3 36486 gidsn 36815 zrdivrng 36816 |
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