Theorem xpima 6173

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 908	. . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ ∨ ¬ (𝐴 ∩ 𝐶) = ∅)
2		df-ima 5664	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3		df-res 5663	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
4	3	rneqi 5919	. . . . . . . 8 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
5	2, 4	eqtri 2784	. . . . . . 7 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6		inxp 5809	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7	6	rneqi 5919	. . . . . . 7 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
8		inv1 4348	. . . . . . . . 9 ⊢ (𝐵 ∩ V) = 𝐵
9	8	xpeq2i 5678	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
10	9	rneqi 5919	. . . . . . 7 ⊢ ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ((𝐴 ∩ 𝐶) × 𝐵)
11	5, 7, 10	3eqtri 2788	. . . . . 6 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × 𝐵)
12		xpeq1 5665	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = (∅ × 𝐵))
13		0xp 5750	. . . . . . . . 9 ⊢ (∅ × 𝐵) = ∅
14	12, 13	eqtrdi 2812	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
15	14	rneqd 5920	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ran ∅)
16		rn0 5908	. . . . . . 7 ⊢ ran ∅ = ∅
17	15, 16	eqtrdi 2812	. . . . . 6 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
18	11, 17	eqtrid 2808	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
19	18	ancli 558	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20		df-ne 2957	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐴 ∩ 𝐶) = ∅)
21		rnxp 6161	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = 𝐵)
22	20, 21	sylbir 238	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = 𝐵)
23	11, 22	eqtrid 2808	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
24	23	ancli 558	. . . 4 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
25	19, 24	orim12i 922	. . 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 4524	. 2 ⊢ (((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) ↔ (((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅) ∨ (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵)))
28	26, 27	mpbir 234	1 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ≠ wne 2956  Vcvv 3451   ∩ cin 3898  ∅c0 4279  ifcif 4482   × cxp 5649  ran crn 5652   ↾ cres 5653   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  xpima1  6174  xpima2  6175  imadifxp  33177  bj-xpimasn  37838  bj-imdirco  38079