Theorem rnsnopg 6221

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 5671	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
2		dfdm4 5884	. . . 4 ⊢ dom {⟨𝐵, 𝐴⟩} = ran ◡{⟨𝐵, 𝐴⟩}
3		df-rn 5671	. . . 4 ⊢ ran ◡{⟨𝐵, 𝐴⟩} = dom ◡◡{⟨𝐵, 𝐴⟩}
4		cnvcnvsn 6219	. . . . 5 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
5	4	dmeqi 5893	. . . 4 ⊢ dom ◡◡{⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
6	2, 3, 5	3eqtri 2789	. . 3 ⊢ dom {⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
7	1, 6	eqtr4i 2788	. 2 ⊢ ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8		dmsnopg 6213	. 2 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
9	7, 8	eqtrid 2809	1 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {csn 4588  ⟨cop 4594  ^◡ccnv 5659  dom cdm 5660  ran crn 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671

This theorem is used by:  rnpropg  6222  rnsnop  6224  funcnvpr  6598  funcnvtp  6599  f1ounsn  7270  dprdsn  20114  noextend  27841  usgr1e  29606  1loopgredg  29862  1egrvtxdg0  29872  uspgrloopedg  29879  rnressnsn  33033  cosnopne  33050  rnsnf  45930