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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 6177 | . 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 1542 ∈ wcel 2107 Vcvv 3447 {csn 4590 ⟨cop 4596 ran crn 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 |
This theorem is referenced by: op2nda 6184 fpr 7104 en1 8971 en1OLD 8972 fodomfi 9275 dcomex 10391 s1rn 14496 axlowdimlem13 27952 ex-rn 29433 ex-ima 29435 ptrest 36127 poimirlem3 36131 gidsn 36461 zrdivrng 36462 |
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