Theorem rnxpss 6192

Description: The range of a Cartesian product is included in its 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 5696	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 6177	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 5915	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 6191	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 4030	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 4030	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3951   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by:  ssxpb  6194  ssrnres  6198  resssxp  6290  funssxp  6764  fconst  6794  dff2  7119  dff3  7120  fliftf  7335  frxp2  8169  frxp3  8176  marypha1lem  9473  marypha1  9474  dfac12lem2  10185  brdom4  10570  nqerf  10970  xptrrel  15019  lern  18636  cnconst2  23291  lmss  23306  tsmsxplem1  24161  causs  25332  i1f0  25722  itg10  25723  taylf  26402  noextendseq  27712  perpln2  28719  gsumpart  33060  locfinref  33840  sitg0  34348  heicant  37662  rntrclfvOAI  42702  rtrclex  43630  trclexi  43633  rtrclexi  43634  cnvtrcl0  43639  rntrcl  43641  brtrclfv2  43740  xphe  43794  rfovcnvf1od  44017