Theorem rnxpm 5215

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 4783	. . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 5204	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4980	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4	1, 3	eqtri 2259	. 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5		dmxpm 5000	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → dom (𝐵 × 𝐴) = 𝐵)
6	4, 5	eqtrid 2283	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1402  ∃wex 1545   ∈ wcel 2209   × cxp 4770  ^◡ccnv 4771  dom cdm 4772  ran crn 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-dm 4782  df-rn 4783

This theorem is referenced by:  ssxpbm  5221  ssxp2  5223  xpexr2m  5227  xpima2m  5233  unixpm  5321  djuexb  7378  exmidfodomrlemim  7547  elply2  15819