Theorem rnxpss 6124

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 5630	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 6109	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 5848	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 6123	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 3977	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 3977	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3898   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630

This theorem is referenced by:  ssxpb  6126  ssrnres  6130  resssxp  6222  funssxp  6684  fconst  6714  dff2  7038  dff3  7039  fliftf  7255  frxp2  8080  frxp3  8087  marypha1lem  9324  marypha1  9325  dfac12lem2  10043  brdom4  10428  nqerf  10828  xptrrel  14889  lern  18499  cnconst2  23199  lmss  23214  tsmsxplem1  24069  causs  25226  i1f0  25616  itg10  25617  taylf  26296  noextendseq  27607  perpln2  28690  gsumpart  33044  locfinref  33875  sitg0  34380  heicant  37715  rntrclfvOAI  42808  rtrclex  43734  trclexi  43737  rtrclexi  43738  cnvtrcl0  43743  rntrcl  43745  brtrclfv2  43844  xphe  43898  rfovcnvf1od  44121