Theorem xpima2m 5051

Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima2m	⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem xpima2m

Step	Hyp	Ref	Expression
1		df-ima 4617	. . . 4 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2		df-res 4616	. . . . 5 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
3	2	rneqi 4832	. . . 4 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4		inxp 4738	. . . . 5 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
5	4	rneqi 4832	. . . 4 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
6	1, 3, 5	3eqtri 2190	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7		rnxpm 5033	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (𝐵 ∩ V))
8	6, 7	syl5eq 2211	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝐴 × 𝐵) “ 𝐶) = (𝐵 ∩ V))
9		inv1 3445	. 2 ⊢ (𝐵 ∩ V) = 𝐵
10	8, 9	eqtrdi 2215	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Syntax hints:   → wi 4   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   ∩ cin 3115   × cxp 4602  ran crn 4605   ↾ cres 4606   “ cima 4607

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  xpimasn  5052