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

Theorem imadif 6569
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 674 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
21exbii 1849 . . . . . . 7 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
3 19.40 1888 . . . . . . 7 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
42, 3sylbi 216 . . . . . 6 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
5 nfv 1916 . . . . . . . . . . 11 𝑥Fun 𝐹
6 nfe1 2146 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)
75, 6nfan 1901 . . . . . . . . . 10 𝑥(Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵))
8 funmo 6500 . . . . . . . . . . . . . 14 (Fun 𝐹 → ∃*𝑥 𝑦𝐹𝑥)
9 vex 3445 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 3445 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
119, 10brcnv 5825 . . . . . . . . . . . . . . 15 (𝑦𝐹𝑥𝑥𝐹𝑦)
1211mobii 2546 . . . . . . . . . . . . . 14 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
138, 12sylib 217 . . . . . . . . . . . . 13 (Fun 𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14 mopick 2625 . . . . . . . . . . . . 13 ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1513, 14sylan 580 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1615con2d 134 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐵 → ¬ 𝑥𝐹𝑦))
17 imnan 400 . . . . . . . . . . 11 ((𝑥𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐵𝑥𝐹𝑦))
1816, 17sylib 217 . . . . . . . . . 10 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
197, 18alrimi 2205 . . . . . . . . 9 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦))
2019ex 413 . . . . . . . 8 (Fun 𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)))
21 exancom 1863 . . . . . . . 8 (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) ↔ ∃𝑥𝑥𝐵𝑥𝐹𝑦))
22 alnex 1782 . . . . . . . 8 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
2320, 21, 223imtr3g 294 . . . . . . 7 (Fun 𝐹 → (∃𝑥𝑥𝐵𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
2423anim2d 612 . . . . . 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 1876 . . . . . . 7 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
2722, 26sylan2br 595 . . . . . 6 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
28 andi 1005 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29 ianor 979 . . . . . . . . 9 (¬ (𝑥𝐵𝑥𝐹𝑦) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦))
3029anbi2i 623 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31 an32 643 . . . . . . . . 9 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵))
32 pm3.24 403 . . . . . . . . . . . 12 ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
3332intnan 487 . . . . . . . . . . 11 ¬ (𝑥𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34 anass 469 . . . . . . . . . . 11 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
3533, 34mtbir 322 . . . . . . . . . 10 ¬ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
3635biorfi 936 . . . . . . . . 9 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3731, 36bitri 274 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3828, 30, 373bitr4i 302 . . . . . . 7 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
3938exbii 1849 . . . . . 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 3908 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4342anbi1i 624 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4443exbii 1849 . . . 4 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
459elima2 6006 . . . . 5 (𝑦 ∈ (𝐹𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
469elima2 6006 . . . . . 6 (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4746notbii 319 . . . . 5 𝑦 ∈ (𝐹𝐵) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4845, 47anbi12i 627 . . . 4 ((𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
4941, 44, 483bitr4g 313 . . 3 (Fun 𝐹 → (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵))))
509elima2 6006 . . 3 (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
51 eldif 3908 . . 3 (𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵)) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)))
5249, 50, 513bitr4g 313 . 2 (Fun 𝐹 → (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵))))
5352eqrdv 2734 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844  wal 1538   = wceq 1540  wex 1780  wcel 2105  ∃*wmo 2536  cdif 3895   class class class wbr 5093  ccnv 5620  cima 5624  Fun wfun 6474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-br 5094  df-opab 5156  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6482
This theorem is referenced by:  imain  6570  resdif  6789  difpreima  6999  domunsncan  8938  phplem2  9074  php3  9078  phplem4OLD  9086  php3OLD  9090  infdifsn  9515  cantnfp1lem3  9538  enfin1ai  10242  fin1a2lem7  10264  symgfixelsi  19140  dprdf1o  19731  frlmlbs  21111  f1lindf  21136  cnclima  22526  iscncl  22527  qtopcld  22971  qtoprest  22975  qtopcmap  22977  mbfimaicc  24902  ismbf3d  24925  i1fd  24952  ballotlemfrc  32793  poimirlem2  35935  poimirlem4  35937  poimirlem6  35939  poimirlem7  35940  poimirlem9  35942  poimirlem11  35944  poimirlem12  35945  poimirlem13  35946  poimirlem14  35947  poimirlem16  35949  poimirlem19  35952  poimirlem23  35956
  Copyright terms: Public domain W3C validator