Theorem rnxpss 4849

Description: The range of a cross product is a subclass of the 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 4439	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 4837	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4625	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 4848	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 3054	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 3054	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 2997   × cxp 4426  ^◡ccnv 4427  dom cdm 4428  ran crn 4429

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439

This theorem is referenced by:  rnxpss2  4851  rnxpid  4852  ssxpbm  4853  ssxp2  4855  ssrnres  4860  funssxp  5165  fconst  5190  dff2  5427  fliftf  5560  tfrcllembfn  6104  frecuzrdgtcl  9784  exmidsbthrlem  11569