Theorem imadifxp 32686

Description: Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.)

Assertion

Ref	Expression
imadifxp	⊢ (𝐶 ⊆ 𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵))

Proof of Theorem imadifxp

Step	Hyp	Ref	Expression
1		ima0 6036	. . . 4 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅) = ∅
2		imaeq2 6015	. . . 4 ⊢ (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅))
3		imaeq2 6015	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝑅 “ 𝐶) = (𝑅 “ ∅))
4		ima0 6036	. . . . . . 7 ⊢ (𝑅 “ ∅) = ∅
5	3, 4	eqtrdi 2788	. . . . . 6 ⊢ (𝐶 = ∅ → (𝑅 “ 𝐶) = ∅)
6	5	difeq1d 4066	. . . . 5 ⊢ (𝐶 = ∅ → ((𝑅 “ 𝐶) ∖ 𝐵) = (∅ ∖ 𝐵))
7		0dif 4346	. . . . 5 ⊢ (∅ ∖ 𝐵) = ∅
8	6, 7	eqtrdi 2788	. . . 4 ⊢ (𝐶 = ∅ → ((𝑅 “ 𝐶) ∖ 𝐵) = ∅)
9	1, 2, 8	3eqtr4a 2798	. . 3 ⊢ (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵))
10	9	adantl 481	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐶 = ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵))
11		uncom 4099	. . . . 5 ⊢ (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅)
12		un0 4335	. . . . 5 ⊢ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)
13	11, 12	eqtr2i 2761	. . . 4 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
14		inundif 4420	. . . . . . . . 9 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) = 𝑅
15	14	imaeq1i 6016	. . . . . . . 8 ⊢ (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (𝑅 “ 𝐶)
16		imaundir 6108	. . . . . . . 8 ⊢ (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
17	15, 16	eqtr3i 2762	. . . . . . 7 ⊢ (𝑅 “ 𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
18	17	difeq1i 4063	. . . . . 6 ⊢ ((𝑅 “ 𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵)
19		difundir 4232	. . . . . 6 ⊢ ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
20	18, 19	eqtri 2760	. . . . 5 ⊢ ((𝑅 “ 𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
21		inss2 4179	. . . . . . . . 9 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
22		imass1 6060	. . . . . . . . 9 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶))
23		ssdif 4085	. . . . . . . . 9 ⊢ (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵))
24	21, 22, 23	mp2b 10	. . . . . . . 8 ⊢ (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵)
25		xpima 6140	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)
26		incom 4150	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
27		dfss2 3908	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶)
28	27	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∩ 𝐴) = 𝐶)
29	26, 28	eqtr3id 2786	. . . . . . . . . . . . . 14 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
30	29	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∩ 𝐶) = 𝐶)
31		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → 𝐶 ≠ ∅)
32	30, 31	eqnetrd 3000	. . . . . . . . . . . 12 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∩ 𝐶) ≠ ∅)
33		neneq 2939	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ¬ (𝐴 ∩ 𝐶) = ∅)
34		iffalse 4476	. . . . . . . . . . . 12 ⊢ (¬ (𝐴 ∩ 𝐶) = ∅ → if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) = 𝐵)
35	32, 33, 34	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) = 𝐵)
36	25, 35	eqtrid 2784	. . . . . . . . . 10 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
37	36	difeq1d 4066	. . . . . . . . 9 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = (𝐵 ∖ 𝐵))
38		difid 4317	. . . . . . . . 9 ⊢ (𝐵 ∖ 𝐵) = ∅
39	37, 38	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = ∅)
40	24, 39	sseqtrid 3965	. . . . . . 7 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅)
41		ss0 4343	. . . . . . 7 ⊢ ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅ → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
42	40, 41	syl 17	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
43		df-ima 5637	. . . . . . . . . . 11 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶)
44		df-res 5636	. . . . . . . . . . . 12 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
45	44	rneqi 5886	. . . . . . . . . . 11 ⊢ ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
46	43, 45	eqtri 2760	. . . . . . . . . 10 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
47	46	ineq1i 4157	. . . . . . . . 9 ⊢ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵)
48		xpss1 5643	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 × V) ⊆ (𝐴 × V))
49		sslin 4184	. . . . . . . . . . 11 ⊢ ((𝐶 × V) ⊆ (𝐴 × V) → ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
50		rnss 5888	. . . . . . . . . . 11 ⊢ (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
51	48, 49, 50	3syl 18	. . . . . . . . . 10 ⊢ (𝐶 ⊆ 𝐴 → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
52		ssn0 4345	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅) → 𝐴 ≠ ∅)
53	52	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → 𝐴 ≠ ∅)
54		inss1 4178	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V)
55		ssdif 4085	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V) → (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵)))
56	54, 55	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵))
57		incom 4150	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V))
58		indif2 4222	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
59	57, 58	eqtr3i 2762	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
60		difxp2 6124	. . . . . . . . . . . . . . 15 ⊢ (𝐴 × (V ∖ 𝐵)) = ((𝐴 × V) ∖ (𝐴 × 𝐵))
61	56, 59, 60	3sstr4i 3974	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵))
62		rnss 5888	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
63	61, 62	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
64		rnxp 6128	. . . . . . . . . . . . 13 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × (V ∖ 𝐵)) = (V ∖ 𝐵))
65	63, 64	sseqtrd 3959	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
66		disj2 4399	. . . . . . . . . . . 12 ⊢ ((ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅ ↔ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
67	65, 66	sylibr 234	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
68	53, 67	syl 17	. . . . . . . . . 10 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
69		ssdisj 4401	. . . . . . . . . 10 ⊢ ((ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∧ (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵) = ∅)
70	51, 68, 69	syl2an2 687	. . . . . . . . 9 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵) = ∅)
71	47, 70	eqtrid 2784	. . . . . . . 8 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅)
72		disj3 4395	. . . . . . . 8 ⊢ ((((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅ ↔ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
73	71, 72	sylib 218	. . . . . . 7 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
74	73	eqcomd 2743	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
75	42, 74	uneq12d 4110	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵)) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
76	20, 75	eqtrid 2784	. . . 4 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → ((𝑅 “ 𝐶) ∖ 𝐵) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
77	13, 76	eqtr4id 2791	. . 3 ⊢ ((𝐶 ≠ ∅ ∧ 𝐶 ⊆ 𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵))
78	77	ancoms 458	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵))
79	10, 78	pm2.61dane 3020	1 ⊢ (𝐶 ⊆ 𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 “ 𝐶) ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ≠ wne 2933  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ifcif 4467   × cxp 5622  ran crn 5625   ↾ cres 5626   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by: (None)