Theorem rnxpm 5100

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 4675	. . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 5089	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4868	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4	1, 3	eqtri 2217	. 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5		dmxpm 4887	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → dom (𝐵 × 𝐴) = 𝐵)
6	4, 5	eqtrid 2241	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1364  ∃wex 1506   ∈ wcel 2167   × cxp 4662  ^◡ccnv 4663  dom cdm 4664  ran crn 4665

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-dm 4674  df-rn 4675

This theorem is referenced by:  ssxpbm  5106  ssxp2  5108  xpexr2m  5112  xpima2m  5118  unixpm  5206  djuexb  7119  exmidfodomrlemim  7280  elply2  15055