Theorem xpima1 5174

Description: The image by a cross product. (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		df-ima 4731	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2		df-res 4730	. . . 4 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
3	2	rneqi 4951	. . 3 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4		inxp 4855	. . . 4 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
5	4	rneqi 4951	. . 3 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
6	1, 3, 5	3eqtri 2254	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7		xpeq1 4732	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8		0xp 4798	. . . 4 ⊢ (∅ × (𝐵 ∩ V)) = ∅
9	7, 8	eqtrdi 2278	. . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅)
10		rneq 4950	. . . 4 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ∅)
11		rn0 4979	. . . 4 ⊢ ran ∅ = ∅
12	10, 11	eqtrdi 2278	. . 3 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅)
13	9, 12	syl 14	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅)
14	6, 13	eqtrid 2274	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1395  Vcvv 2799   ∩ cin 3196  ∅c0 3491   × cxp 4716  ran crn 4719   ↾ cres 4720   “ cima 4721

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731

This theorem is referenced by: (None)