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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imadifxp Structured version   Visualization version   GIF version

Theorem imadifxp 32099
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))

Proof of Theorem imadifxp
StepHypRef Expression
1 ima0 6075 . . . 4 ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅) = ∅
2 imaeq2 6054 . . . 4 (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) “ ∅))
3 imaeq2 6054 . . . . . . 7 (𝐶 = ∅ → (𝑅𝐶) = (𝑅 “ ∅))
4 ima0 6075 . . . . . . 7 (𝑅 “ ∅) = ∅
53, 4eqtrdi 2786 . . . . . 6 (𝐶 = ∅ → (𝑅𝐶) = ∅)
65difeq1d 4120 . . . . 5 (𝐶 = ∅ → ((𝑅𝐶) ∖ 𝐵) = (∅ ∖ 𝐵))
7 0dif 4400 . . . . 5 (∅ ∖ 𝐵) = ∅
86, 7eqtrdi 2786 . . . 4 (𝐶 = ∅ → ((𝑅𝐶) ∖ 𝐵) = ∅)
91, 2, 83eqtr4a 2796 . . 3 (𝐶 = ∅ → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
109adantl 480 . 2 ((𝐶𝐴𝐶 = ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
11 uncom 4152 . . . . 5 (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅)
12 un0 4389 . . . . 5 (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∪ ∅) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)
1311, 12eqtr2i 2759 . . . 4 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
14 inundif 4477 . . . . . . . . 9 ((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) = 𝑅
1514imaeq1i 6055 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (𝑅𝐶)
16 imaundir 6149 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) ∪ (𝑅 ∖ (𝐴 × 𝐵))) “ 𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
1715, 16eqtr3i 2760 . . . . . . 7 (𝑅𝐶) = (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
1817difeq1i 4117 . . . . . 6 ((𝑅𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵)
19 difundir 4279 . . . . . 6 ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
2018, 19eqtri 2758 . . . . 5 ((𝑅𝐶) ∖ 𝐵) = ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
21 inss2 4228 . . . . . . . . 9 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
22 imass1 6099 . . . . . . . . 9 ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶))
23 ssdif 4138 . . . . . . . . 9 (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ⊆ ((𝐴 × 𝐵) “ 𝐶) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵))
2421, 22, 23mp2b 10 . . . . . . . 8 (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵)
25 xpima 6180 . . . . . . . . . . 11 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
26 incom 4200 . . . . . . . . . . . . . . 15 (𝐶𝐴) = (𝐴𝐶)
27 df-ss 3964 . . . . . . . . . . . . . . . 16 (𝐶𝐴 ↔ (𝐶𝐴) = 𝐶)
2827biimpi 215 . . . . . . . . . . . . . . 15 (𝐶𝐴 → (𝐶𝐴) = 𝐶)
2926, 28eqtr3id 2784 . . . . . . . . . . . . . 14 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
3029adantl 480 . . . . . . . . . . . . 13 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (𝐴𝐶) = 𝐶)
31 simpl 481 . . . . . . . . . . . . 13 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → 𝐶 ≠ ∅)
3230, 31eqnetrd 3006 . . . . . . . . . . . 12 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (𝐴𝐶) ≠ ∅)
33 neneq 2944 . . . . . . . . . . . 12 ((𝐴𝐶) ≠ ∅ → ¬ (𝐴𝐶) = ∅)
34 iffalse 4536 . . . . . . . . . . . 12 (¬ (𝐴𝐶) = ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
3532, 33, 343syl 18 . . . . . . . . . . 11 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
3625, 35eqtrid 2782 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
3736difeq1d 4120 . . . . . . . . 9 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = (𝐵𝐵))
38 difid 4369 . . . . . . . . 9 (𝐵𝐵) = ∅
3937, 38eqtrdi 2786 . . . . . . . 8 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝐴 × 𝐵) “ 𝐶) ∖ 𝐵) = ∅)
4024, 39sseqtrid 4033 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅)
41 ss0 4397 . . . . . . 7 ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ⊆ ∅ → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
4240, 41syl 17 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ∅)
43 df-ima 5688 . . . . . . . . . . 11 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶)
44 df-res 5687 . . . . . . . . . . . 12 ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4544rneqi 5935 . . . . . . . . . . 11 ran ((𝑅 ∖ (𝐴 × 𝐵)) ↾ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4643, 45eqtri 2758 . . . . . . . . . 10 ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V))
4746ineq1i 4207 . . . . . . . . 9 (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵)
48 xpss1 5694 . . . . . . . . . . 11 (𝐶𝐴 → (𝐶 × V) ⊆ (𝐴 × V))
49 sslin 4233 . . . . . . . . . . 11 ((𝐶 × V) ⊆ (𝐴 × V) → ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
50 rnss 5937 . . . . . . . . . . 11 (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
5148, 49, 503syl 18 . . . . . . . . . 10 (𝐶𝐴 → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ⊆ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)))
52 ssn0 4399 . . . . . . . . . . . 12 ((𝐶𝐴𝐶 ≠ ∅) → 𝐴 ≠ ∅)
5352ancoms 457 . . . . . . . . . . 11 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → 𝐴 ≠ ∅)
54 inss1 4227 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V)
55 ssdif 4138 . . . . . . . . . . . . . . . 16 (((𝐴 × V) ∩ 𝑅) ⊆ (𝐴 × V) → (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵)))
5654, 55ax-mp 5 . . . . . . . . . . . . . . 15 (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵)) ⊆ ((𝐴 × V) ∖ (𝐴 × 𝐵))
57 incom 4200 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V))
58 indif2 4269 . . . . . . . . . . . . . . . 16 ((𝐴 × V) ∩ (𝑅 ∖ (𝐴 × 𝐵))) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
5957, 58eqtr3i 2760 . . . . . . . . . . . . . . 15 ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) = (((𝐴 × V) ∩ 𝑅) ∖ (𝐴 × 𝐵))
60 difxp2 6164 . . . . . . . . . . . . . . 15 (𝐴 × (V ∖ 𝐵)) = ((𝐴 × V) ∖ (𝐴 × 𝐵))
6156, 59, 603sstr4i 4024 . . . . . . . . . . . . . 14 ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵))
62 rnss 5937 . . . . . . . . . . . . . 14 (((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (𝐴 × (V ∖ 𝐵)) → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
6361, 62mp1i 13 . . . . . . . . . . . . 13 (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ ran (𝐴 × (V ∖ 𝐵)))
64 rnxp 6168 . . . . . . . . . . . . 13 (𝐴 ≠ ∅ → ran (𝐴 × (V ∖ 𝐵)) = (V ∖ 𝐵))
6563, 64sseqtrd 4021 . . . . . . . . . . . 12 (𝐴 ≠ ∅ → ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
66 disj2 4456 . . . . . . . . . . . 12 ((ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅ ↔ ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ⊆ (V ∖ 𝐵))
6765, 66sylibr 233 . . . . . . . . . . 11 (𝐴 ≠ ∅ → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
6853, 67syl 17 . . . . . . . . . 10 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐴 × V)) ∩ 𝐵) = ∅)
69 ssdisj 4458 . . . . . . . . . 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) = ∅)
7051, 68, 69syl2an2 682 . . . . . . . . 9 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (ran ((𝑅 ∖ (𝐴 × 𝐵)) ∩ (𝐶 × V)) ∩ 𝐵) = ∅)
7147, 70eqtrid 2782 . . . . . . . 8 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅)
72 disj3 4452 . . . . . . . 8 ((((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∩ 𝐵) = ∅ ↔ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
7371, 72sylib 217 . . . . . . 7 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵))
7473eqcomd 2736 . . . . . 6 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) = ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶))
7542, 74uneq12d 4163 . . . . 5 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((((𝑅 ∩ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵) ∪ (((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) ∖ 𝐵)) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
7620, 75eqtrid 2782 . . . 4 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅𝐶) ∖ 𝐵) = (∅ ∪ ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶)))
7713, 76eqtr4id 2789 . . 3 ((𝐶 ≠ ∅ ∧ 𝐶𝐴) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
7877ancoms 457 . 2 ((𝐶𝐴𝐶 ≠ ∅) → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
7910, 78pm2.61dane 3027 1 (𝐶𝐴 → ((𝑅 ∖ (𝐴 × 𝐵)) “ 𝐶) = ((𝑅𝐶) ∖ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wne 2938  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  ifcif 4527   × cxp 5673  ran crn 5676  cres 5677  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by: (None)
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