Theorem rnxpss2 4879

Description: Upper bound for the range of a binary relation. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
rnxpss2	|- ( R C_ ( A X. B ) -> ran R C_ B )

Proof of Theorem rnxpss2

Step	Hyp	Ref	Expression
1		rnss 4680	. 2 |- ( R C_ ( A X. B ) -> ran R C_ ran ( A X. B ) )
2		rnxpss 4877	. 2 |- ran ( A X. B ) C_ B
3	1, 2	syl6ss 3040	1 |- ( R C_ ( A X. B ) -> ran R C_ B )

Syntax hints:   -> wi 4   C_ wss 3002   X. cxp 4452   ran crn 4455

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-cnv 4462  df-dm 4464  df-rn 4465

This theorem is referenced by:  cossxp2  4969