Theorem rnxp 2657

Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37.

Assertion

Ref	Expression
rnxp	⊢ (¬ A = ∅ → ran (A × B) = B)

Proof of Theorem rnxp

Step	Hyp	Ref	Expression
1		dmxp 2552	. 2 ⊢ (¬ A = ∅ → dom (B × A) = B)
2		df-rn 2429	. . 3 ⊢ ran (A × B) = dom ◡(A × B)
3		cnvxp 2651	. . . 4 ⊢ ◡(A × B) = (B × A)
4	3	dmeqi 2532	. . 3 ⊢ dom ◡(A × B) = dom (B × A)
5	2, 4	eqtr 1119	. 2 ⊢ ran (A × B) = dom (B × A)
6	1, 5	syl5eq 1136	1 ⊢ (¬ A = ∅ → ran (A × B) = B)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091  ∅c0 1707   × cxp 2408  ^◡ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  aceq5lem3 3560  fodomb 3615

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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