Theorem rnxpss 5193

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 4759	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 5180	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4956	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 5192	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 3269	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 3269	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3210   × cxp 4746  ^◡ccnv 4747  dom cdm 4748  ran crn 4749

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-dm 4758  df-rn 4759

This theorem is referenced by:  rnxpss2  5195  rnxpid  5196  ssxpbm  5197  ssxp2  5199  ssrnres  5204  funssxp  5531  fconst  5562  dff2  5820  fliftf  5971  tfrcllembfn  6587  frecuzrdgtcl  10770  cnconst2  15085  lmss  15098  exmidsbthrlem  16789