Theorem xpima1 4787

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 4376	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2		df-res 4375	. . . 4 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
3	2	rneqi 4580	. . 3 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4		inxp 4488	. . . 4 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
5	4	rneqi 4580	. . 3 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
6	1, 3, 5	3eqtri 2105	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7		xpeq1 4377	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8		0xp 4438	. . . 4 ⊢ (∅ × (𝐵 ∩ V)) = ∅
9	7, 8	syl6eq 2129	. . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅)
10		rneq 4579	. . . 4 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ∅)
11		rn0 4606	. . . 4 ⊢ ran ∅ = ∅
12	10, 11	syl6eq 2129	. . 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	syl5eq 2125	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1284  Vcvv 2601   ∩ cin 2972  ∅c0 3251   × cxp 4361  ran crn 4364   ↾ cres 4365   “ cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by: (None)