Theorem xpima2m 5184

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 4738	. . . 4 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
2		df-res 4737	. . . . 5 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
3	2	rneqi 4960	. . . 4 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
4		inxp 4864	. . . . 5 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
5	4	rneqi 4960	. . . 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 5166	. . 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 3531	. 2 ⊢ (𝐵 ∩ V) = 𝐵
10	8, 9	eqtrdi 2280	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ 𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802   ∩ cin 3199   × cxp 4723  ran crn 4726   ↾ cres 4727   “ cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  xpimasn  5185