Theorem rnxpss 6162

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 5678	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 6147	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 5895	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 6161	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 4009	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 4009	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3941   × cxp 5665  ^◡ccnv 5666  dom cdm 5667  ran crn 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678

This theorem is referenced by:  ssxpb  6164  ssrnres  6168  resssxp  6260  funssxp  6737  fconst  6768  dff2  7091  dff3  7092  fliftf  7305  frxp2  8125  frxp3  8132  marypha1lem  9425  marypha1  9426  dfac12lem2  10136  brdom4  10522  nqerf  10922  xptrrel  14929  lern  18552  cnconst2  23131  lmss  23146  tsmsxplem1  24001  causs  25170  i1f0  25560  itg10  25561  taylf  26238  noextendseq  27541  perpln2  28456  gsumpart  32701  locfinref  33341  sitg0  33865  heicant  37027  rntrclfvOAI  41981  rtrclex  42918  trclexi  42921  rtrclexi  42922  cnvtrcl0  42927  rntrcl  42929  brtrclfv2  43028  xphe  43082  rfovcnvf1od  43305