Theorem rnxpss 5042

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 4622	. 2 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 5029	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4812	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4		dmxpss 5041	. . 3 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
5	3, 4	eqsstri 3179	. 2 ⊢ dom ◡(𝐴 × 𝐵) ⊆ 𝐵
6	1, 5	eqsstri 3179	1 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵

Syntax hints:   ⊆ wss 3121   × cxp 4609  ^◡ccnv 4610  dom cdm 4611  ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  rnxpss2  5044  rnxpid  5045  ssxpbm  5046  ssxp2  5048  ssrnres  5053  funssxp  5367  fconst  5393  dff2  5640  fliftf  5778  tfrcllembfn  6336  frecuzrdgtcl  10368  cnconst2  13027  lmss  13040  exmidsbthrlem  14054