Theorem rnsnopg 6230

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 5693	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
2		dfdm4 5902	. . . 4 ⊢ dom {⟨𝐵, 𝐴⟩} = ran ◡{⟨𝐵, 𝐴⟩}
3		df-rn 5693	. . . 4 ⊢ ran ◡{⟨𝐵, 𝐴⟩} = dom ◡◡{⟨𝐵, 𝐴⟩}
4		cnvcnvsn 6228	. . . . 5 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
5	4	dmeqi 5911	. . . 4 ⊢ dom ◡◡{⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
6	2, 3, 5	3eqtri 2760	. . 3 ⊢ dom {⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
7	1, 6	eqtr4i 2759	. 2 ⊢ ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8		dmsnopg 6222	. 2 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
9	7, 8	eqtrid 2780	1 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {csn 4632  ⟨cop 4638  ^◡ccnv 5681  dom cdm 5682  ran crn 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693

This theorem is referenced by:  rnpropg  6231  rnsnop  6233  funcnvpr  6620  funcnvtp  6621  dprdsn  20000  noextend  27619  usgr1e  29078  1loopgredg  29335  1egrvtxdg0  29345  uspgrloopedg  29352  cosnopne  32495  rnsnf  44587