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 5444 . . . 4 ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅) = ∅
2 imaeq2 5425 . . . 4 (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅))
3 imaeq2 5425 . . . . . . 7 (𝐶 = ∅ → (𝑅𝐶) = (𝑅 “ ∅))
4 ima0 5444 . . . . . . 7 (𝑅 “ ∅) = ∅
53, 4syl6eq 2676 . . . . . 6 (𝐶 = ∅ → (𝑅𝐶) = ∅)
65difeq1d 3710 . . . . 5 (𝐶 = ∅ → ((𝑅𝐶) ∖ 𝐵) = (∅ ∖ 𝐵))
7 0dif 3954 . . . . 5 (∅ ∖ 𝐵) = ∅
86, 7syl6eq 2676 . . . 4 (𝐶 = ∅ → ((𝑅𝐶) ∖ 𝐵) = ∅)
91, 2, 83eqtr4a 2686 . . 3 (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
109adantl 482 . 2 ((𝐶𝐴𝐶 = ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
11 inundif 4023 . . . . . . . . 9 ((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) = 𝑅
1211imaeq1i 5426 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (𝑅𝐶)
13 imaundir 5509 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
1412, 13eqtr3i 2650 . . . . . . 7 (𝑅𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
1514difeq1i 3707 . . . . . 6 ((𝑅𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵)
16 difundir 3861 . . . . . 6 ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
1715, 16eqtri 2648 . . . . 5 ((𝑅𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
18 inss2 3817 . . . . . . . . 9 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
19 imass1 5463 . . . . . . . . 9 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶))
20 ssdif 3728 . . . . . . . . 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 5539 . . . . . . . . . . 11 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
23 incom 3788 . . . . . . . . . . . . . . 15 (𝐶𝐴) = (𝐴𝐶)
24 df-ss 3574 . . . . . . . . . . . . . . . 16 (𝐶𝐴 ↔ (𝐶𝐴) = 𝐶)
2524biimpi 206 . . . . . . . . . . . . . . 15 (𝐶𝐴 → (𝐶𝐴) = 𝐶)
2623, 25syl5eqr 2674 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
2726adantl 482 . . . . . . . . . . . . 13 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (𝐴𝐶) = 𝐶)
28 simpl 473 . . . . . . . . . . . . 13 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → 𝐶 ≠ ∅)
2927, 28eqnetrd 2863 . . . . . . . . . . . 12 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (𝐴𝐶) ≠ ∅)
30 df-ne 2797 . . . . . . . . . . . . 13 ((𝐴𝐶) ≠ ∅ ↔ ¬ (𝐴𝐶) = ∅)
3130biimpi 206 . . . . . . . . . . . 12 ((𝐴𝐶) ≠ ∅ → ¬ (𝐴𝐶) = ∅)
32 iffalse 4072 . . . . . . . . . . . 12 (¬ (𝐴𝐶) = ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
3329, 31, 323syl 18 . . . . . . . . . . 11 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
3422, 33syl5eq 2672 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
3534difeq1d 3710 . . . . . . . . 9 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = (𝐵𝐵))
36 difid 3927 . . . . . . . . 9 (𝐵𝐵) = ∅
3735, 36syl6eq 2676 . . . . . . . 8 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = ∅)
3821, 37syl5sseq 3637 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅)
39 ss0 3951 . . . . . . 7 ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅ → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
4038, 39syl 17 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
41 df-ima 5092 . . . . . . . . . . 11 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶)
42 df-res 5091 . . . . . . . . . . . 12 ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4342rneqi 5316 . . . . . . . . . . 11 ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4441, 43eqtri 2648 . . . . . . . . . 10 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4544ineq1i 3793 . . . . . . . . 9 (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵)
46 xpss1 5194 . . . . . . . . . . . 12 (𝐶𝐴 → (𝐶 × V) ⊆ (𝐴 × V))
47 sslin 3822 . . . . . . . . . . . 12 ((𝐶 × V) ⊆ (𝐴 × V) → ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
48 rnss 5318 . . . . . . . . . . . 12 (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
4946, 47, 483syl 18 . . . . . . . . . . 11 (𝐶𝐴 → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
5049adantl 482 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
51 ssn0 3953 . . . . . . . . . . . 12 ((𝐶𝐴𝐶 ≠ ∅) → 𝐴 ≠ ∅)
5251ancoms 469 . . . . . . . . . . 11 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → 𝐴 ≠ ∅)
53 inss1 3816 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V)
54 ssdif 3728 . . . . . . . . . . . . . . . 16 (((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V) → (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵)))
5553, 54ax-mp 5 . . . . . . . . . . . . . . 15 (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵))
56 incom 3788 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V))
57 indif2 3851 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
5856, 57eqtr3i 2650 . . . . . . . . . . . . . . 15 ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
59 difxp2 5523 . . . . . . . . . . . . . . 15 (𝐴 × (V ∖ 𝐵)) = ((𝐴 × V) ∖ (𝐴 × 𝐵))
6055, 58, 593sstr4i 3628 . . . . . . . . . . . . . 14 ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵))
61 rnss 5318 . . . . . . . . . . . . . 14 (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
6260, 61mp1i 13 . . . . . . . . . . . . 13 (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
63 rnxp 5527 . . . . . . . . . . . . 13 (𝐴 ≠ ∅ → ran (𝐴 × (V ∖ 𝐵)) = (V ∖ 𝐵))
6462, 63sseqtrd 3625 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
65 disj2 4001 . . . . . . . . . . . 12 ((ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅ ↔ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
6664, 65sylibr 224 . . . . . . . . . . 11 (𝐴 ≠ ∅ → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
6752, 66syl 17 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
68 ssdisj 4003 . . . . . . . . . 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) = ∅)
6950, 67, 68syl2anc 692 . . . . . . . . 9 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵) = ∅)
7045, 69syl5eq 2672 . . . . . . . 8 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅)
71 disj3 3998 . . . . . . . 8 ((((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅ ↔ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
7270, 71sylib 208 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
7372eqcomd 2632 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
7440, 73uneq12d 3751 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵)) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
7517, 74syl5eq 2672 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅𝐶) ∖ 𝐵) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
76 uncom 3740 . . . . 5 (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅)
77 un0 3944 . . . . 5 (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)
7876, 77eqtr2i 2649 . . . 4 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
7975, 78syl6reqr 2679 . . 3 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
8079ancoms 469 . 2 ((𝐶𝐴𝐶 ≠ ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
8110, 80pm2.61dane 2883 1 (𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1480   ≠ wne 2796  Vcvv 3191   ∖ cdif 3557   ∪ cun 3558   ∩ cin 3559   ⊆ wss 3560  ∅c0 3896  ifcif 4063   × cxp 5077  ran crn 5080   ↾ cres 5081   “ cima 5082 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator