Theorem rnxpss 4697

Description: The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
rnxpss	⊢ ran (A × B) ⊆ B

Proof of Theorem rnxpss

Step	Hyp	Ref	Expression
1		df-rn 4299	. 2 ⊢ ran (A × B) = dom ◡(A × B)
2		cnvxp 4685	. . . 4 ⊢ ◡(A × B) = (B × A)
3	2	dmeqi 4479	. . 3 ⊢ dom ◡(A × B) = dom (B × A)
4		dmxpss 4696	. . 3 ⊢ dom (B × A) ⊆ B
5	3, 4	eqsstri 2969	. 2 ⊢ dom ◡(A × B) ⊆ B
6	1, 5	eqsstri 2969	1 ⊢ ran (A × B) ⊆ B

Syntax hints:   ⊆ wss 2911   × cxp 4286  ^◡ccnv 4287  dom cdm 4288  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  rnxpid  4698  ssxpbm  4699  ssxp2  4701  ssrnres  4706  funssxp  5003  fconst  5025  dff2  5254  fliftf  5382