Theorem rnxpss 6170

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 5672	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 6154	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 5894	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 6169	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 3982	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 3982	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3904   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  ssxpb  6172  ssrnres  6176  resssxp  6271  funssxp  6734  fconst  6764  dff2  7094  dff3  7095  fliftf  7313  frxp2  8139  frxp3  8146  marypha1lem  9392  marypha1  9393  dfac12lem2  10127  brdom4  10513  nqerf  10914  xptrrel  15017  lern  18646  cnconst2  23419  lmss  23434  tsmsxplem1  24289  causs  25436  i1f0  25825  itg10  25826  taylf  26500  noextendseq  27807  perpln2  28966  gsumpart  33349  locfinref  34197  sitg0  34702  heicant  38272  rntrclfvOAI  43392  rtrclex  44313  trclexi  44316  rtrclexi  44317  cnvtrcl0  44322  rntrcl  44324  brtrclfv2  44423  xphe  44477  rfovcnvf1od  44700