![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5530 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = dom ◡{〈𝐴, 𝐵〉} | |
2 | dfdm4 5728 | . . . 4 ⊢ dom {〈𝐵, 𝐴〉} = ran ◡{〈𝐵, 𝐴〉} | |
3 | df-rn 5530 | . . . 4 ⊢ ran ◡{〈𝐵, 𝐴〉} = dom ◡◡{〈𝐵, 𝐴〉} | |
4 | cnvcnvsn 6043 | . . . . 5 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
5 | 4 | dmeqi 5737 | . . . 4 ⊢ dom ◡◡{〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
6 | 2, 3, 5 | 3eqtri 2825 | . . 3 ⊢ dom {〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
7 | 1, 6 | eqtr4i 2824 | . 2 ⊢ ran {〈𝐴, 𝐵〉} = dom {〈𝐵, 𝐴〉} |
8 | dmsnopg 6037 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝐵, 𝐴〉} = {𝐵}) | |
9 | 7, 8 | syl5eq 2845 | 1 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {csn 4525 〈cop 4531 ◡ccnv 5518 dom cdm 5519 ran crn 5520 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: rnpropg 6046 rnsnop 6048 funcnvpr 6386 funcnvtp 6387 dprdsn 19151 usgr1e 27035 1loopgredg 27291 1egrvtxdg0 27301 uspgrloopedg 27308 cosnopne 30454 noextend 33286 rnsnf 41810 |
Copyright terms: Public domain | W3C validator |