Theorem xpima 6129

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 5629	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3		df-res 5628	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
4	3	rneqi 5877	. . . . . . . 8 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
5	2, 4	eqtri 2754	. . . . . . 7 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6		inxp 5771	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7	6	rneqi 5877	. . . . . . 7 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
8		inv1 4348	. . . . . . . . 9 ⊢ (𝐵 ∩ V) = 𝐵
9	8	xpeq2i 5643	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
10	9	rneqi 5877	. . . . . . 7 ⊢ ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ((𝐴 ∩ 𝐶) × 𝐵)
11	5, 7, 10	3eqtri 2758	. . . . . 6 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × 𝐵)
12		xpeq1 5630	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = (∅ × 𝐵))
13		0xp 5715	. . . . . . . . 9 ⊢ (∅ × 𝐵) = ∅
14	12, 13	eqtrdi 2782	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
15	14	rneqd 5878	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ran ∅)
16		rn0 5866	. . . . . . 7 ⊢ ran ∅ = ∅
17	15, 16	eqtrdi 2782	. . . . . 6 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
18	11, 17	eqtrid 2778	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
19	18	ancli 548	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20		df-ne 2929	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐴 ∩ 𝐶) = ∅)
21		rnxp 6117	. . . . . . 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 2778	. . . . 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 4517	. 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 1541   ≠ wne 2928  Vcvv 3436   ∩ cin 3901  ∅c0 4283  ifcif 4475   × cxp 5614  ran crn 5617   ↾ cres 5618   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  xpima1  6130  xpima2  6131  imadifxp  32576  bj-xpimasn  36988  bj-imdirco  37223