Theorem rnxpss 6171

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 5687	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 6156	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 5904	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 6170	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 4016	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 4016	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3948   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ran crn 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by:  ssxpb  6173  ssrnres  6177  resssxp  6269  funssxp  6746  fconst  6777  dff2  7100  dff3  7101  fliftf  7311  frxp2  8129  frxp3  8136  marypha1lem  9427  marypha1  9428  dfac12lem2  10138  brdom4  10524  nqerf  10924  xptrrel  14926  lern  18543  cnconst2  22786  lmss  22801  tsmsxplem1  23656  causs  24814  i1f0  25203  itg10  25204  taylf  25872  noextendseq  27167  perpln2  27959  gsumpart  32202  locfinref  32816  sitg0  33340  heicant  36518  rntrclfvOAI  41419  rtrclex  42358  trclexi  42361  rtrclexi  42362  cnvtrcl0  42367  rntrcl  42369  brtrclfv2  42468  xphe  42522  rfovcnvf1od  42745