Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rnsnopg | 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 4609 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = dom ◡{〈𝐴, 𝐵〉} | |
2 | dfdm4 4790 | . . . 4 ⊢ dom {〈𝐵, 𝐴〉} = ran ◡{〈𝐵, 𝐴〉} | |
3 | df-rn 4609 | . . . 4 ⊢ ran ◡{〈𝐵, 𝐴〉} = dom ◡◡{〈𝐵, 𝐴〉} | |
4 | cnvcnvsn 5074 | . . . . 5 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
5 | 4 | dmeqi 4799 | . . . 4 ⊢ dom ◡◡{〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
6 | 2, 3, 5 | 3eqtri 2189 | . . 3 ⊢ dom {〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
7 | 1, 6 | eqtr4i 2188 | . 2 ⊢ ran {〈𝐴, 𝐵〉} = dom {〈𝐵, 𝐴〉} |
8 | dmsnopg 5069 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝐵, 𝐴〉} = {𝐵}) | |
9 | 7, 8 | syl5eq 2209 | 1 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 {csn 3570 〈cop 3573 ◡ccnv 4597 dom cdm 4598 ran crn 4599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: rnpropg 5077 rnsnop 5078 fprg 5662 |
Copyright terms: Public domain | W3C validator |