Theorem rnsnopg 5107

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 4637	. . 3 |- ran { <. A , B >. } = dom `' { <. A , B >. }
2		dfdm4 4819	. . . 4 |- dom { <. B , A >. } = ran `' { <. B , A >. }
3		df-rn 4637	. . . 4 |- ran `' { <. B , A >. } = dom `' `' { <. B , A >. }
4		cnvcnvsn 5105	. . . . 5 |- `' `' { <. B , A >. } = `' { <. A , B >. }
5	4	dmeqi 4828	. . . 4 |- dom `' `' { <. B , A >. } = dom `' { <. A , B >. }
6	2, 3, 5	3eqtri 2202	. . 3 |- dom { <. B , A >. } = dom `' { <. A , B >. }
7	1, 6	eqtr4i 2201	. 2 |- ran { <. A , B >. } = dom { <. B , A >. }
8		dmsnopg 5100	. 2 |- ( A e. V -> dom { <. B , A >. } = { B } )
9	7, 8	eqtrid 2222	1 |- ( A e. V -> ran { <. A , B >. } = { B } )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {csn 3592   <.cop 3595   `'ccnv 4625   dom cdm 4626   ran crn 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637

This theorem is referenced by:  rnpropg  5108  rnsnop  5109  fprg  5699