Theorem rnxpm 5060

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 4639	. . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵)
2		cnvxp 5049	. . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
3	2	dmeqi 4830	. . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴)
4	1, 3	eqtri 2198	. 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴)
5		dmxpm 4849	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → dom (𝐵 × 𝐴) = 𝐵)
6	4, 5	eqtrid 2222	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1353  ∃wex 1492   ∈ wcel 2148   × cxp 4626  ^◡ccnv 4627  dom cdm 4628  ran crn 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  ssxpbm  5066  ssxp2  5068  xpexr2m  5072  xpima2m  5078  unixpm  5166  djuexb  7045  exmidfodomrlemim  7202