Theorem xpima1 6108

Description: Direct image by a Cartesian product (case of empty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima1	⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Proof of Theorem xpima1

Step	Hyp	Ref	Expression
1		xpima 6107	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)
2		iftrue 4477	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) = ∅)
3	1, 2	eqtrid 2789	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3896  ∅c0 4267  ifcif 4471   × cxp 5605   “ cima 5610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-br 5088  df-opab 5150  df-xp 5613  df-rel 5614  df-cnv 5615  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620

This theorem is referenced by:  bj-xpima1snALT  35203  arearect  41250