Theorem rnsnopg 5213

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 e. V -> ran { <. A , B >. } = { B } )

Proof of Theorem rnsnopg

Step	Hyp	Ref	Expression
1		df-rn 4734	. . 3 |- ran { <. A , B >. } = dom `' { <. A , B >. }
2		dfdm4 4921	. . . 4 |- dom { <. B , A >. } = ran `' { <. B , A >. }
3		df-rn 4734	. . . 4 |- ran `' { <. B , A >. } = dom `' `' { <. B , A >. }
4		cnvcnvsn 5211	. . . . 5 |- `' `' { <. B , A >. } = `' { <. A , B >. }
5	4	dmeqi 4930	. . . 4 |- dom `' `' { <. B , A >. } = dom `' { <. A , B >. }
6	2, 3, 5	3eqtri 2254	. . 3 |- dom { <. B , A >. } = dom `' { <. A , B >. }
7	1, 6	eqtr4i 2253	. 2 |- ran { <. A , B >. } = dom { <. B , A >. }
8		dmsnopg 5206	. 2 |- ( A e. V -> dom { <. B , A >. } = { B } )
9	7, 8	eqtrid 2274	1 |- ( A e. V -> ran { <. A , B >. } = { B } )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {csn 3667   <.cop 3670   `'ccnv 4722   dom cdm 4723   ran crn 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  rnpropg  5214  rnsnop  5215  fprg  5832  usgr1e  16080  1loopgredg  16110