MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imadif Structured version   Visualization version   GIF version

Theorem imadif 6586
Description: The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
Assertion
Ref Expression
imadif (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))

Proof of Theorem imadif
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anandir 676 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
21exbii 1851 . . . . . . 7 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
3 19.40 1890 . . . . . . 7 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
42, 3sylbi 216 . . . . . 6 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
5 nfv 1918 . . . . . . . . . . 11 𝑥Fun 𝐹
6 nfe1 2148 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)
75, 6nfan 1903 . . . . . . . . . 10 𝑥(Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵))
8 funmo 6517 . . . . . . . . . . . . . 14 (Fun 𝐹 → ∃*𝑥 𝑦𝐹𝑥)
9 vex 3448 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 3448 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
119, 10brcnv 5839 . . . . . . . . . . . . . . 15 (𝑦𝐹𝑥𝑥𝐹𝑦)
1211mobii 2543 . . . . . . . . . . . . . 14 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
138, 12sylib 217 . . . . . . . . . . . . 13 (Fun 𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14 mopick 2622 . . . . . . . . . . . . 13 ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1513, 14sylan 581 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1615con2d 134 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐵 → ¬ 𝑥𝐹𝑦))
17 imnan 401 . . . . . . . . . . 11 ((𝑥𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐵𝑥𝐹𝑦))
1816, 17sylib 217 . . . . . . . . . 10 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
197, 18alrimi 2207 . . . . . . . . 9 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦))
2019ex 414 . . . . . . . 8 (Fun 𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)))
21 exancom 1865 . . . . . . . 8 (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) ↔ ∃𝑥𝑥𝐵𝑥𝐹𝑦))
22 alnex 1784 . . . . . . . 8 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
2320, 21, 223imtr3g 295 . . . . . . 7 (Fun 𝐹 → (∃𝑥𝑥𝐵𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
2423anim2d 613 . . . . . 6 (Fun 𝐹 → ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
254, 24syl5 34 . . . . 5 (Fun 𝐹 → (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
26 19.29r 1878 . . . . . . 7 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
2722, 26sylan2br 596 . . . . . 6 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
28 andi 1007 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29 ianor 981 . . . . . . . . 9 (¬ (𝑥𝐵𝑥𝐹𝑦) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦))
3029anbi2i 624 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31 an32 645 . . . . . . . . 9 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵))
32 pm3.24 404 . . . . . . . . . . . 12 ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
3332intnan 488 . . . . . . . . . . 11 ¬ (𝑥𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34 anass 470 . . . . . . . . . . 11 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
3533, 34mtbir 323 . . . . . . . . . 10 ¬ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
3635biorfi 938 . . . . . . . . 9 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3731, 36bitri 275 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3828, 30, 373bitr4i 303 . . . . . . 7 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
3938exbii 1851 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4027, 39sylib 217 . . . . 5 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4125, 40impbid1 224 . . . 4 (Fun 𝐹 → (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
42 eldif 3921 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4342anbi1i 625 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4443exbii 1851 . . . 4 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
459elima2 6020 . . . . 5 (𝑦 ∈ (𝐹𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
469elima2 6020 . . . . . 6 (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4746notbii 320 . . . . 5 𝑦 ∈ (𝐹𝐵) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4845, 47anbi12i 628 . . . 4 ((𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
4941, 44, 483bitr4g 314 . . 3 (Fun 𝐹 → (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵))))
509elima2 6020 . . 3 (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
51 eldif 3921 . . 3 (𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵)) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)))
5249, 50, 513bitr4g 314 . 2 (Fun 𝐹 → (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵))))
5352eqrdv 2731 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃*wmo 2533  cdif 3908   class class class wbr 5106  ccnv 5633  cima 5637  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499
This theorem is referenced by:  imain  6587  resdif  6806  difpreima  7016  domunsncan  9019  phplem2  9155  php3  9159  phplem4OLD  9167  php3OLD  9171  infdifsn  9598  cantnfp1lem3  9621  enfin1ai  10325  fin1a2lem7  10347  symgfixelsi  19222  dprdf1o  19816  frlmlbs  21219  f1lindf  21244  cnclima  22635  iscncl  22636  qtopcld  23080  qtoprest  23084  qtopcmap  23086  mbfimaicc  25011  ismbf3d  25034  i1fd  25061  ballotlemfrc  33183  poimirlem2  36126  poimirlem4  36128  poimirlem6  36130  poimirlem7  36131  poimirlem9  36133  poimirlem11  36135  poimirlem12  36136  poimirlem13  36137  poimirlem14  36138  poimirlem16  36140  poimirlem19  36143  poimirlem23  36147
  Copyright terms: Public domain W3C validator