Theorem rnsnopg 6226

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 5676	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
2		dfdm4 5889	. . . 4 ⊢ dom {⟨𝐵, 𝐴⟩} = ran ◡{⟨𝐵, 𝐴⟩}
3		df-rn 5676	. . . 4 ⊢ ran ◡{⟨𝐵, 𝐴⟩} = dom ◡◡{⟨𝐵, 𝐴⟩}
4		cnvcnvsn 6224	. . . . 5 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
5	4	dmeqi 5898	. . . 4 ⊢ dom ◡◡{⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
6	2, 3, 5	3eqtri 2797	. . 3 ⊢ dom {⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
7	1, 6	eqtr4i 2796	. 2 ⊢ ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8		dmsnopg 6218	. 2 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
9	7, 8	eqtrid 2817	1 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2150  {csn 4594  ⟨cop 4600  ^◡ccnv 5664  dom cdm 5665  ran crn 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by:  rnpropg  6227  rnsnop  6229  funcnvpr  6602  funcnvtp  6603  f1ounsn  7274  dprdsn  20111  noextend  27810  usgr1e  29565  1loopgredg  29821  1egrvtxdg0  29831  uspgrloopedg  29838  rnressnsn  32992  cosnopne  33009  rnsnf  45854