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| 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 6241 | . 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 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ran crn 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 | 
| This theorem is referenced by: op2nda 6248 fpr 7174 en1 9064 fodomfi 9350 fodomfiOLD 9370 dcomex 10487 s1rn 14637 axlowdimlem13 28969 ex-rn 30459 ex-ima 30461 ptrest 37626 poimirlem3 37630 gidsn 37959 zrdivrng 37960 | 
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