Theorem rnpropg 5216

Description: The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Assertion

Ref	Expression
rnpropg	|- ( ( A e. V /\ B e. W ) -> ran { <. A , C >. , <. B , D >. } = { C , D } )

Proof of Theorem rnpropg

Step	Hyp	Ref	Expression
1		df-pr 3676	. . 3 |- { <. A , C >. , <. B , D >. } = ( { <. A , C >. } u. { <. B , D >. } )
2	1	rneqi 4960	. 2 |- ran { <. A , C >. , <. B , D >. } = ran ( { <. A , C >. } u. { <. B , D >. } )
3		rnsnopg 5215	. . . . 5 |- ( A e. V -> ran { <. A , C >. } = { C } )
4	3	adantr 276	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran { <. A , C >. } = { C } )
5		rnsnopg 5215	. . . . 5 |- ( B e. W -> ran { <. B , D >. } = { D } )
6	5	adantl 277	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran { <. B , D >. } = { D } )
7	4, 6	uneq12d 3362	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ran { <. A , C >. } u. ran { <. B , D >. } ) = ( { C } u. { D } ) )
8		rnun 5145	. . 3 |- ran ( { <. A , C >. } u. { <. B , D >. } ) = ( ran { <. A , C >. } u. ran { <. B , D >. } )
9		df-pr 3676	. . 3 |- { C , D } = ( { C } u. { D } )
10	7, 8, 9	3eqtr4g 2289	. 2 |- ( ( A e. V /\ B e. W ) -> ran ( { <. A , C >. } u. { <. B , D >. } ) = { C , D } )
11	2, 10	eqtrid 2276	1 |- ( ( A e. V /\ B e. W ) -> ran { <. A , C >. , <. B , D >. } = { C , D } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   u. cun 3198   {csn 3669   {cpr 3670   <.cop 3672   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)