Theorem rnsnopg 6210

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 5665	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
2		dfdm4 5875	. . . 4 ⊢ dom {⟨𝐵, 𝐴⟩} = ran ◡{⟨𝐵, 𝐴⟩}
3		df-rn 5665	. . . 4 ⊢ ran ◡{⟨𝐵, 𝐴⟩} = dom ◡◡{⟨𝐵, 𝐴⟩}
4		cnvcnvsn 6208	. . . . 5 ⊢ ◡◡{⟨𝐵, 𝐴⟩} = ◡{⟨𝐴, 𝐵⟩}
5	4	dmeqi 5884	. . . 4 ⊢ dom ◡◡{⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
6	2, 3, 5	3eqtri 2762	. . 3 ⊢ dom {⟨𝐵, 𝐴⟩} = dom ◡{⟨𝐴, 𝐵⟩}
7	1, 6	eqtr4i 2761	. 2 ⊢ ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐵, 𝐴⟩}
8		dmsnopg 6202	. 2 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝐵, 𝐴⟩} = {𝐵})
9	7, 8	eqtrid 2782	1 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {csn 4601  ⟨cop 4607  ^◡ccnv 5653  dom cdm 5654  ran crn 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by:  rnpropg  6211  rnsnop  6213  funcnvpr  6598  funcnvtp  6599  f1ounsn  7265  dprdsn  20019  noextend  27630  usgr1e  29224  1loopgredg  29481  1egrvtxdg0  29491  uspgrloopedg  29498  cosnopne  32671  rnsnf  45208