Theorem rnsnopg 5180

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 4704	. . 3 |- ran { <. A , B >. } = dom `' { <. A , B >. }
2		dfdm4 4889	. . . 4 |- dom { <. B , A >. } = ran `' { <. B , A >. }
3		df-rn 4704	. . . 4 |- ran `' { <. B , A >. } = dom `' `' { <. B , A >. }
4		cnvcnvsn 5178	. . . . 5 |- `' `' { <. B , A >. } = `' { <. A , B >. }
5	4	dmeqi 4898	. . . 4 |- dom `' `' { <. B , A >. } = dom `' { <. A , B >. }
6	2, 3, 5	3eqtri 2232	. . 3 |- dom { <. B , A >. } = dom `' { <. A , B >. }
7	1, 6	eqtr4i 2231	. 2 |- ran { <. A , B >. } = dom { <. B , A >. }
8		dmsnopg 5173	. 2 |- ( A e. V -> dom { <. B , A >. } = { B } )
9	7, 8	eqtrid 2252	1 |- ( A e. V -> ran { <. A , B >. } = { B } )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   {csn 3643   <.cop 3646   `'ccnv 4692   dom cdm 4693   ran crn 4694

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  rnpropg  5181  rnsnop  5182  fprg  5790