Theorem xpima2m 5191

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 4744	. . . 4 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2		df-res 4743	. . . . 5 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
3	2	rneqi 4966	. . . 4 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4		inxp 4870	. . . . 5 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
5	4	rneqi 4966	. . . 4 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
6	1, 3, 5	3eqtri 2256	. . 3 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7		rnxpm 5173	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = (𝐵 ∩ V))
8	6, 7	eqtrid 2276	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝐴 × 𝐵) “ 𝐶) = (𝐵 ∩ V))
9		inv1 3533	. 2 ⊢ (𝐵 ∩ V) = 𝐵
10	8, 9	eqtrdi 2280	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Syntax hints:   → wi 4   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803   ∩ cin 3200   × cxp 4729  ran crn 4732   ↾ cres 4733   “ cima 4734

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  xpimasn  5192