Theorem xpima1 4993

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 4560	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2		df-res 4559	. . . 4 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
3	2	rneqi 4775	. . 3 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4		inxp 4681	. . . 4 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
5	4	rneqi 4775	. . 3 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
6	1, 3, 5	3eqtri 2165	. 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7		xpeq1 4561	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (∅ × (𝐵 ∩ V)))
8		0xp 4627	. . . 4 ⊢ (∅ × (𝐵 ∩ V)) = ∅
9	7, 8	eqtrdi 2189	. . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅)
10		rneq 4774	. . . 4 ⊢ (((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ∅ → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ∅)
11		rn0 4803	. . . 4 ⊢ ran ∅ = ∅
12	10, 11	eqtrdi 2189	. . 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 2185	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1332  Vcvv 2689   ∩ cin 3075  ∅c0 3368   × cxp 4545  ran crn 4548   ↾ cres 4549   “ cima 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by: (None)