Theorem rnxpss2 5057

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 4852	. 2 |- ( R C_ ( A X. B ) -> ran R C_ ran ( A X. B ) )
2		rnxpss 5055	. 2 |- ran ( A X. B ) C_ B
3	1, 2	sstrdi 3167	1 |- ( R C_ ( A X. B ) -> ran R C_ B )

Syntax hints:   -> wi 4   C_ wss 3129   X. cxp 4620   ran crn 4623

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 4118  ax-pow 4171  ax-pr 4205

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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633

This theorem is referenced by:  cossxp2  5147