Theorem rnxpm 5158

Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with nonempty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
rnxpm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rnxpm

Step	Hyp	Ref	Expression
1		df-rn 4730	. . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 5147	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4924	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4	1, 3	eqtri 2250	. 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5		dmxpm 4944	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → dom (𝐵 × 𝐴) = 𝐵)
6	4, 5	eqtrid 2274	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1395  ∃wex 1538   ∈ wcel 2200   × cxp 4717  ^◡ccnv 4718  dom cdm 4719  ran crn 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  ssxpbm  5164  ssxp2  5166  xpexr2m  5170  xpima2m  5176  unixpm  5264  djuexb  7219  exmidfodomrlemim  7387  elply2  15417