Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imadifxp Structured version   Visualization version   GIF version

 Description: Image of the difference with a Cartesian product. (Contributed by Thierry Arnoux, 13-Dec-2017.)
Assertion
Ref Expression
imadifxp (𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))

StepHypRef Expression
1 ima0 5942 . . . 4 ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅) = ∅
2 imaeq2 5922 . . . 4 (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅))
3 imaeq2 5922 . . . . . . 7 (𝐶 = ∅ → (𝑅𝐶) = (𝑅 “ ∅))
4 ima0 5942 . . . . . . 7 (𝑅 “ ∅) = ∅
53, 4syl6eq 2876 . . . . . 6 (𝐶 = ∅ → (𝑅𝐶) = ∅)
65difeq1d 4101 . . . . 5 (𝐶 = ∅ → ((𝑅𝐶) ∖ 𝐵) = (∅ ∖ 𝐵))
7 0dif 4358 . . . . 5 (∅ ∖ 𝐵) = ∅
86, 7syl6eq 2876 . . . 4 (𝐶 = ∅ → ((𝑅𝐶) ∖ 𝐵) = ∅)
91, 2, 83eqtr4a 2886 . . 3 (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
109adantl 482 . 2 ((𝐶𝐴𝐶 = ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
11 inundif 4429 . . . . . . . . 9 ((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) = 𝑅
1211imaeq1i 5923 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (𝑅𝐶)
13 imaundir 6006 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
1412, 13eqtr3i 2850 . . . . . . 7 (𝑅𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
1514difeq1i 4098 . . . . . 6 ((𝑅𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵)
16 difundir 4260 . . . . . 6 ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
1715, 16eqtri 2848 . . . . 5 ((𝑅𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
18 inss2 4209 . . . . . . . . 9 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
19 imass1 5961 . . . . . . . . 9 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶))
20 ssdif 4119 . . . . . . . . 9 (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵))
2118, 19, 20mp2b 10 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵)
22 xpima 6036 . . . . . . . . . . 11 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
23 incom 4181 . . . . . . . . . . . . . . 15 (𝐶𝐴) = (𝐴𝐶)
24 df-ss 3955 . . . . . . . . . . . . . . . 16 (𝐶𝐴 ↔ (𝐶𝐴) = 𝐶)
2524biimpi 217 . . . . . . . . . . . . . . 15 (𝐶𝐴 → (𝐶𝐴) = 𝐶)
2623, 25syl5eqr 2874 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
2726adantl 482 . . . . . . . . . . . . 13 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (𝐴𝐶) = 𝐶)
28 simpl 483 . . . . . . . . . . . . 13 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → 𝐶 ≠ ∅)
2927, 28eqnetrd 3087 . . . . . . . . . . . 12 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (𝐴𝐶) ≠ ∅)
30 neneq 3026 . . . . . . . . . . . 12 ((𝐴𝐶) ≠ ∅ → ¬ (𝐴𝐶) = ∅)
31 iffalse 4478 . . . . . . . . . . . 12 (¬ (𝐴𝐶) = ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
3229, 30, 313syl 18 . . . . . . . . . . 11 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
3322, 32syl5eq 2872 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
3433difeq1d 4101 . . . . . . . . 9 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = (𝐵𝐵))
35 difid 4333 . . . . . . . . 9 (𝐵𝐵) = ∅
3634, 35syl6eq 2876 . . . . . . . 8 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = ∅)
3721, 36sseqtrid 4022 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅)
38 ss0 4355 . . . . . . 7 ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅ → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
3937, 38syl 17 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
40 df-ima 5566 . . . . . . . . . . 11 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶)
41 df-res 5565 . . . . . . . . . . . 12 ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4241rneqi 5805 . . . . . . . . . . 11 ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4340, 42eqtri 2848 . . . . . . . . . 10 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4443ineq1i 4188 . . . . . . . . 9 (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵)
45 xpss1 5572 . . . . . . . . . . 11 (𝐶𝐴 → (𝐶 × V) ⊆ (𝐴 × V))
46 sslin 4214 . . . . . . . . . . 11 ((𝐶 × V) ⊆ (𝐴 × V) → ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
47 rnss 5807 . . . . . . . . . . 11 (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
4845, 46, 473syl 18 . . . . . . . . . 10 (𝐶𝐴 → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
49 ssn0 4357 . . . . . . . . . . . 12 ((𝐶𝐴𝐶 ≠ ∅) → 𝐴 ≠ ∅)
5049ancoms 459 . . . . . . . . . . 11 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → 𝐴 ≠ ∅)
51 inss1 4208 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V)
52 ssdif 4119 . . . . . . . . . . . . . . . 16 (((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V) → (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵)))
5351, 52ax-mp 5 . . . . . . . . . . . . . . 15 (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵))
54 incom 4181 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V))
55 indif2 4250 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
5654, 55eqtr3i 2850 . . . . . . . . . . . . . . 15 ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
57 difxp2 6020 . . . . . . . . . . . . . . 15 (𝐴 × (V ∖ 𝐵)) = ((𝐴 × V) ∖ (𝐴 × 𝐵))
5853, 56, 573sstr4i 4013 . . . . . . . . . . . . . 14 ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵))
59 rnss 5807 . . . . . . . . . . . . . 14 (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
6058, 59mp1i 13 . . . . . . . . . . . . 13 (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
61 rnxp 6024 . . . . . . . . . . . . 13 (𝐴 ≠ ∅ → ran (𝐴 × (V ∖ 𝐵)) = (V ∖ 𝐵))
6260, 61sseqtrd 4010 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
63 disj2 4409 . . . . . . . . . . . 12 ((ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅ ↔ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
6462, 63sylibr 235 . . . . . . . . . . 11 (𝐴 ≠ ∅ → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
6550, 64syl 17 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
66 ssdisj 4411 . . . . . . . . . 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) = ∅)
6748, 65, 66syl2an2 682 . . . . . . . . 9 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵) = ∅)
6844, 67syl5eq 2872 . . . . . . . 8 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅)
69 disj3 4405 . . . . . . . 8 ((((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅ ↔ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
7068, 69sylib 219 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
7170eqcomd 2830 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
7239, 71uneq12d 4143 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵)) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
7317, 72syl5eq 2872 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅𝐶) ∖ 𝐵) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
74 uncom 4132 . . . . 5 (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅)
75 un0 4347 . . . . 5 (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)
7674, 75eqtr2i 2849 . . . 4 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
7773, 76syl6reqr 2879 . . 3 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
7877ancoms 459 . 2 ((𝐶𝐴𝐶 ≠ ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
7910, 78pm2.61dane 3108 1 (𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530   ≠ wne 3020  Vcvv 3499   ∖ cdif 3936   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  ifcif 4469   × cxp 5551  ran crn 5554   ↾ cres 5555   “ cima 5556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566 This theorem is referenced by: (None)
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