Theorem rnxpss 6161

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 5665	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 6146	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 5884	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 6160	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 4005	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 4005	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3926   × cxp 5652  ^◡ccnv 5653  dom cdm 5654  ran crn 5655

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by:  ssxpb  6163  ssrnres  6167  resssxp  6259  funssxp  6734  fconst  6764  dff2  7089  dff3  7090  fliftf  7308  frxp2  8143  frxp3  8150  marypha1lem  9445  marypha1  9446  dfac12lem2  10159  brdom4  10544  nqerf  10944  xptrrel  14999  lern  18601  cnconst2  23221  lmss  23236  tsmsxplem1  24091  causs  25250  i1f0  25640  itg10  25641  taylf  26320  noextendseq  27631  perpln2  28690  gsumpart  33051  locfinref  33872  sitg0  34378  heicant  37679  rntrclfvOAI  42714  rtrclex  43641  trclexi  43644  rtrclexi  43645  cnvtrcl0  43650  rntrcl  43652  brtrclfv2  43751  xphe  43805  rfovcnvf1od  44028