Theorem rnsnopg 5145

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.)

Assertion

Ref	Expression
rnsnopg	⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Proof of Theorem rnsnopg

Step	Hyp	Ref	Expression
1		df-rn 4671	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
2		dfdm4 4855	. . . 4 ⊢ dom {⟨𝐵, 𝐴⟩} = ran ◡{⟨𝐵, 𝐴⟩}
3		df-rn 4671	. . . 4 ⊢ ran ◡{⟨𝐵, 𝐴⟩} = dom ◡◡{⟨𝐵, 𝐴⟩}
4		cnvcnvsn 5143	. . . . 5 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
5	4	dmeqi 4864	. . . 4 ⊢ dom ◡◡{⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
6	2, 3, 5	3eqtri 2218	. . 3 ⊢ dom {⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
7	1, 6	eqtr4i 2217	. 2 ⊢ ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8		dmsnopg 5138	. 2 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
9	7, 8	eqtrid 2238	1 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {csn 3619  ⟨cop 3622  ^◡ccnv 4659  dom cdm 4660  ran crn 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by:  rnpropg  5146  rnsnop  5147  fprg  5742