Theorem xpima 6171

Description: Direct image by a Cartesian product. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Assertion

Ref	Expression
xpima	⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)

Proof of Theorem xpima

Step	Hyp	Ref	Expression
1		exmid 894	. . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ ∨ ¬ (𝐴 ∩ 𝐶) = ∅)
2		df-ima 5667	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3		df-res 5666	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
4	3	rneqi 5917	. . . . . . . 8 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
5	2, 4	eqtri 2758	. . . . . . 7 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6		inxp 5811	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7	6	rneqi 5917	. . . . . . 7 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
8		inv1 4373	. . . . . . . . 9 ⊢ (𝐵 ∩ V) = 𝐵
9	8	xpeq2i 5681	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
10	9	rneqi 5917	. . . . . . 7 ⊢ ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ((𝐴 ∩ 𝐶) × 𝐵)
11	5, 7, 10	3eqtri 2762	. . . . . 6 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × 𝐵)
12		xpeq1 5668	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = (∅ × 𝐵))
13		0xp 5753	. . . . . . . . 9 ⊢ (∅ × 𝐵) = ∅
14	12, 13	eqtrdi 2786	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
15	14	rneqd 5918	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ran ∅)
16		rn0 5905	. . . . . . 7 ⊢ ran ∅ = ∅
17	15, 16	eqtrdi 2786	. . . . . 6 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
18	11, 17	eqtrid 2782	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
19	18	ancli 548	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20		df-ne 2933	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐴 ∩ 𝐶) = ∅)
21		rnxp 6159	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = 𝐵)
22	20, 21	sylbir 235	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = 𝐵)
23	11, 22	eqtrid 2782	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
24	23	ancli 548	. . . 4 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
25	19, 24	orim12i 908	. . 3 ⊢ (((𝐴 ∩ 𝐶) = ∅ ∨ ¬ (𝐴 ∩ 𝐶) = ∅) → (((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
26	1, 25	ax-mp 5	. 2 ⊢ (((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
27		eqif 4542	. 2 ⊢ (((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
28	26, 27	mpbir 231	1 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ≠ wne 2932  Vcvv 3459   ∩ cin 3925  ∅c0 4308  ifcif 4500   × cxp 5652  ran crn 5655   ↾ cres 5656   “ cima 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667

This theorem is referenced by:  xpima1  6172  xpima2  6173  imadifxp  32582  bj-xpimasn  36973  bj-imdirco  37208