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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, 30-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| rnsnopg | ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rn 5696 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = dom ◡{〈𝐴, 𝐵〉} | |
| 2 | dfdm4 5906 | . . . 4 ⊢ dom {〈𝐵, 𝐴〉} = ran ◡{〈𝐵, 𝐴〉} | |
| 3 | df-rn 5696 | . . . 4 ⊢ ran ◡{〈𝐵, 𝐴〉} = dom ◡◡{〈𝐵, 𝐴〉} | |
| 4 | cnvcnvsn 6239 | . . . . 5 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
| 5 | 4 | dmeqi 5915 | . . . 4 ⊢ dom ◡◡{〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} | 
| 6 | 2, 3, 5 | 3eqtri 2769 | . . 3 ⊢ dom {〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} | 
| 7 | 1, 6 | eqtr4i 2768 | . 2 ⊢ ran {〈𝐴, 𝐵〉} = dom {〈𝐵, 𝐴〉} | 
| 8 | dmsnopg 6233 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝐵, 𝐴〉} = {𝐵}) | |
| 9 | 7, 8 | eqtrid 2789 | 1 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 ◡ccnv 5684 dom cdm 5685 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: rnpropg 6242 rnsnop 6244 funcnvpr 6628 funcnvtp 6629 f1ounsn 7292 dprdsn 20056 noextend 27711 usgr1e 29262 1loopgredg 29519 1egrvtxdg0 29529 uspgrloopedg 29536 cosnopne 32703 rnsnf 45189 | 
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