Theorem xpima 6182

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 907	. . 3 ⊢ ((𝐴 ∩ 𝐶) = ∅ ∨ ¬ (𝐴 ∩ 𝐶) = ∅)
2		df-ima 5676	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ↾ 𝐶)
3		df-res 5675	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V))
4	3	rneqi 5929	. . . . . . . 8 ⊢ ran ((𝐴 × 𝐵) ↾ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
5	2, 4	eqtri 2786	. . . . . . 7 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 × 𝐵) ∩ (𝐶 × V))
6		inxp 5820	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
7	6	rneqi 5929	. . . . . . 7 ⊢ ran ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V))
8		inv1 4356	. . . . . . . . 9 ⊢ (𝐵 ∩ V) = 𝐵
9	8	xpeq2i 5690	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵)
10	9	rneqi 5929	. . . . . . 7 ⊢ ran ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ran ((𝐴 ∩ 𝐶) × 𝐵)
11	5, 7, 10	3eqtri 2790	. . . . . 6 ⊢ ((𝐴 × 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐶) × 𝐵)
12		xpeq1 5677	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = (∅ × 𝐵))
13		0xp 5762	. . . . . . . . 9 ⊢ (∅ × 𝐵) = ∅
14	12, 13	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
15	14	rneqd 5930	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ran ∅)
16		rn0 5918	. . . . . . 7 ⊢ ran ∅ = ∅
17	15, 16	eqtrdi 2814	. . . . . 6 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ran ((𝐴 ∩ 𝐶) × 𝐵) = ∅)
18	11, 17	eqtrid 2810	. . . . 5 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
19	18	ancli 557	. . . 4 ⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = ∅))
20		df-ne 2959	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ ↔ ¬ (𝐴 ∩ 𝐶) = ∅)
21		rnxp 6170	. . . . . . 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 2810	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
24	23	ancli 557	. . . 4 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → (¬ (𝐴 ∩ 𝐶) = ∅ ∧ ((𝐴 × 𝐵) “ 𝐶) = 𝐵))
25	19, 24	orim12i 921	. . 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 4530	. 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)

Syntax hints:  ¬ wn 3   ∧ wa 400   ∨ wo 860   = wceq 1570   ≠ wne 2958  Vcvv 3455   ∩ cin 3905  ∅c0 4287  ifcif 4488   × cxp 5661  ran crn 5664   ↾ cres 5665   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  xpima1  6183  xpima2  6184  imadifxp  32924  bj-xpimasn  37569  bj-imdirco  37812