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Theorem imadif 6308
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 673 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
21exbii 1829 . . . . . . 7 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
3 19.40 1868 . . . . . . 7 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
42, 3sylbi 218 . . . . . 6 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
5 nfv 1892 . . . . . . . . . . 11 𝑥Fun 𝐹
6 nfe1 2120 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)
75, 6nfan 1881 . . . . . . . . . 10 𝑥(Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵))
8 funmo 6241 . . . . . . . . . . . . . 14 (Fun 𝐹 → ∃*𝑥 𝑦𝐹𝑥)
9 vex 3440 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 3440 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
119, 10brcnv 5639 . . . . . . . . . . . . . . 15 (𝑦𝐹𝑥𝑥𝐹𝑦)
1211mobii 2586 . . . . . . . . . . . . . 14 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
138, 12sylib 219 . . . . . . . . . . . . 13 (Fun 𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14 mopick 2680 . . . . . . . . . . . . 13 ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1513, 14sylan 580 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1615con2d 136 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐵 → ¬ 𝑥𝐹𝑦))
17 imnan 400 . . . . . . . . . . 11 ((𝑥𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐵𝑥𝐹𝑦))
1816, 17sylib 219 . . . . . . . . . 10 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
197, 18alrimi 2178 . . . . . . . . 9 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦))
2019ex 413 . . . . . . . 8 (Fun 𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)))
21 exancom 1842 . . . . . . . 8 (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) ↔ ∃𝑥𝑥𝐵𝑥𝐹𝑦))
22 alnex 1763 . . . . . . . 8 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
2320, 21, 223imtr3g 296 . . . . . . 7 (Fun 𝐹 → (∃𝑥𝑥𝐵𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
2423anim2d 611 . . . . . 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 1856 . . . . . . 7 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
2722, 26sylan2br 594 . . . . . 6 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)))
28 andi 1002 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
29 ianor 976 . . . . . . . . 9 (¬ (𝑥𝐵𝑥𝐹𝑦) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦))
3029anbi2i 622 . . . . . . . 8 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐹𝑦)))
31 an32 642 . . . . . . . . 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 324 . . . . . . . . . 10 ¬ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)
3635biorfi 933 . . . . . . . . 9 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3731, 36bitri 276 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ 𝑥𝐹𝑦)))
3828, 30, 373bitr4i 304 . . . . . . 7 (((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
3938exbii 1829 . . . . . 6 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ ¬ (𝑥𝐵𝑥𝐹𝑦)) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4027, 39sylib 219 . . . . 5 ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)) → ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4125, 40impbid1 226 . . . 4 (Fun 𝐹 → (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
42 eldif 3869 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
4342anbi1i 623 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
4443exbii 1829 . . . 4 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
459elima2 5812 . . . . 5 (𝑦 ∈ (𝐹𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
469elima2 5812 . . . . . 6 (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4746notbii 321 . . . . 5 𝑦 ∈ (𝐹𝐵) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
4845, 47anbi12i 626 . . . 4 ((𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
4941, 44, 483bitr4g 315 . . 3 (Fun 𝐹 → (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵))))
509elima2 5812 . . 3 (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
51 eldif 3869 . . 3 (𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵)) ↔ (𝑦 ∈ (𝐹𝐴) ∧ ¬ 𝑦 ∈ (𝐹𝐵)))
5249, 50, 513bitr4g 315 . 2 (Fun 𝐹 → (𝑦 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑦 ∈ ((𝐹𝐴) ∖ (𝐹𝐵))))
5352eqrdv 2793 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 842  wal 1520   = wceq 1522  wex 1761  wcel 2081  ∃*wmo 2574  cdif 3856   class class class wbr 4962  ccnv 5442  cima 5446  Fun wfun 6219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-fun 6227
This theorem is referenced by:  imain  6309  resdif  6503  difpreima  6700  domunsncan  8464  phplem4  8546  php3  8550  infdifsn  8966  cantnfp1lem3  8989  enfin1ai  9652  fin1a2lem7  9674  symgfixelsi  18294  dprdf1o  18871  frlmlbs  20623  f1lindf  20648  cnclima  21560  iscncl  21561  qtopcld  22005  qtoprest  22009  qtopcmap  22011  mbfimaicc  23915  ismbf3d  23938  i1fd  23965  ballotlemfrc  31401  poimirlem2  34425  poimirlem4  34427  poimirlem6  34429  poimirlem7  34430  poimirlem9  34432  poimirlem11  34434  poimirlem12  34435  poimirlem13  34436  poimirlem14  34437  poimirlem16  34439  poimirlem19  34442  poimirlem23  34446
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