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Theorem imadif 6011
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 889 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
21exbii 1814 . . . . . . 7 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
3 19.40 1837 . . . . . . 7 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
42, 3sylbi 207 . . . . . 6 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
5 nfv 1883 . . . . . . . . . . 11 𝑥Fun 𝐹
6 nfe1 2067 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)
75, 6nfan 1868 . . . . . . . . . 10 𝑥(Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵))
8 funmo 5942 . . . . . . . . . . . . . 14 (Fun 𝐹 → ∃*𝑥 𝑦𝐹𝑥)
9 vex 3234 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 3234 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
119, 10brcnv 5337 . . . . . . . . . . . . . . 15 (𝑦𝐹𝑥𝑥𝐹𝑦)
1211mobii 2521 . . . . . . . . . . . . . 14 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
138, 12sylib 208 . . . . . . . . . . . . 13 (Fun 𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14 mopick 2564 . . . . . . . . . . . . 13 ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1513, 14sylan 487 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1615con2d 129 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐵 → ¬ 𝑥𝐹𝑦))
17 imnan 437 . . . . . . . . . . 11 ((𝑥𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐵𝑥𝐹𝑦))
1816, 17sylib 208 . . . . . . . . . 10 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
197, 18alrimi 2120 . . . . . . . . 9 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦))
2019ex 449 . . . . . . . 8 (Fun 𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)))
21 exancom 1827 . . . . . . . 8 (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) ↔ ∃𝑥𝑥𝐵𝑥𝐹𝑦))
22 alnex 1746 . . . . . . . 8 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
2320, 21, 223imtr3g 284 . . . . . . 7 (Fun 𝐹 → (∃𝑥𝑥𝐵𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
2423anim2d 588 . . . . . 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 1842 . . . . . . 7 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
2722, 26sylan2br 492 . . . . . 6 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
28 andi 929 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29 ianor 508 . . . . . . . . 9 (¬ (𝑥𝐵𝑥𝐹𝑦) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦))
3029anbi2i 730 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31 an32 856 . . . . . . . . 9 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵))
32 pm3.24 944 . . . . . . . . . . . 12 ¬ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)
3332intnan 980 . . . . . . . . . . 11 ¬ (𝑥𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦))
34 anass 682 . . . . . . . . . . 11 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦) ↔ (𝑥𝐴 ∧ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐹𝑦)))
3533, 34mtbir 312 . . . . . . . . . 10 ¬ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
3635biorfi 421 . . . . . . . . 9 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3731, 36bitri 264 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3828, 30, 373bitr4i 292 . . . . . . 7 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
3938exbii 1814 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4027, 39sylib 208 . . . . 5 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4125, 40impbid1 215 . . . 4 (Fun 𝐹 → (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
42 eldif 3617 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4342anbi1i 731 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4443exbii 1814 . . . 4 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
459elima2 5507 . . . . 5 (𝑦 ∈ (𝐹𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
469elima2 5507 . . . . . 6 (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4746notbii 309 . . . . 5 𝑦 ∈ (𝐹𝐵) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4845, 47anbi12i 733 . . . 4 ((𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
4941, 44, 483bitr4g 303 . . 3 (Fun 𝐹 → (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵))))
509elima2 5507 . . 3 (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
51 eldif 3617 . . 3 (𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵)) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)))
5249, 50, 513bitr4g 303 . 2 (Fun 𝐹 → (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵))))
5352eqrdv 2649 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃*wmo 2499  cdif 3604   class class class wbr 4685  ccnv 5142  cima 5146  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928
This theorem is referenced by:  imain  6012  resdif  6195  difpreima  6383  domunsncan  8101  phplem4  8183  php3  8187  infdifsn  8592  cantnfp1lem3  8615  enfin1ai  9244  fin1a2lem7  9266  symgfixelsi  17901  dprdf1o  18477  frlmlbs  20184  f1lindf  20209  cnclima  21120  iscncl  21121  qtopcld  21564  qtoprest  21568  qtopcmap  21570  mbfimaicc  23445  ismbf3d  23466  i1fd  23493  ballotlemfrc  30716  poimirlem2  33541  poimirlem4  33543  poimirlem6  33545  poimirlem7  33546  poimirlem9  33548  poimirlem11  33550  poimirlem12  33551  poimirlem13  33552  poimirlem14  33553  poimirlem16  33555  poimirlem19  33558  poimirlem23  33562
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