Theorem rnsnop 2637

Description: The range of a singleton of an ordered pair is the singleton of the second member.

Hypotheses

Ref	Expression
cnvsn.1	⊢ A ∈ V
cnvsn.2	⊢ B ∈ V

Assertion

Ref	Expression
rnsnop	⊢ ran {⟨A, B⟩} = {B}

Proof of Theorem rnsnop

Step	Hyp	Ref	Expression
1		df-rn 2429	. 2 ⊢ ran {⟨A, B⟩} = dom ◡{⟨A, B⟩}
2		cnvsn.1	. . . . 5 ⊢ A ∈ V
3		cnvsn.2	. . . . 5 ⊢ B ∈ V
4	2, 3	cnvsn 2636	. . . 4 ⊢ ◡{⟨A, B⟩} = {⟨B, A⟩}
5	4	dmeqi 2532	. . 3 ⊢ dom ◡{⟨A, B⟩} = dom {⟨B, A⟩}
6		dmsnop 2547	. . 3 ⊢ dom {⟨B, A⟩} = {B}
7	5, 6	eqtr 1119	. 2 ⊢ dom ◡{⟨A, B⟩} = {B}
8	1, 7	eqtr 1119	1 ⊢ ran {⟨A, B⟩} = {B}

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ^◡ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  en1 3331

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

metamath.org