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Mirrors > Home > MPE Home > Th. List > rnsnopg | 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, 30-Apr-2015.) |
Ref | Expression |
---|---|
rnsnopg | ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5600 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = dom ◡{〈𝐴, 𝐵〉} | |
2 | dfdm4 5804 | . . . 4 ⊢ dom {〈𝐵, 𝐴〉} = ran ◡{〈𝐵, 𝐴〉} | |
3 | df-rn 5600 | . . . 4 ⊢ ran ◡{〈𝐵, 𝐴〉} = dom ◡◡{〈𝐵, 𝐴〉} | |
4 | cnvcnvsn 6122 | . . . . 5 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
5 | 4 | dmeqi 5813 | . . . 4 ⊢ dom ◡◡{〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
6 | 2, 3, 5 | 3eqtri 2770 | . . 3 ⊢ dom {〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
7 | 1, 6 | eqtr4i 2769 | . 2 ⊢ ran {〈𝐴, 𝐵〉} = dom {〈𝐵, 𝐴〉} |
8 | dmsnopg 6116 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝐵, 𝐴〉} = {𝐵}) | |
9 | 7, 8 | eqtrid 2790 | 1 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {csn 4561 〈cop 4567 ◡ccnv 5588 dom cdm 5589 ran crn 5590 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 |
This theorem is referenced by: rnpropg 6125 rnsnop 6127 funcnvpr 6496 funcnvtp 6497 dprdsn 19639 usgr1e 27612 1loopgredg 27868 1egrvtxdg0 27878 uspgrloopedg 27885 cosnopne 31027 noextend 33869 rnsnf 42721 |
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