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Theorem imadif 4922
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 525 . . . . . . . 8 (((x A ¬ x B) x𝐹y) ↔ ((x A x𝐹y) x B x𝐹y)))
21exbii 1493 . . . . . . 7 (x((x A ¬ x B) x𝐹y) ↔ x((x A x𝐹y) x B x𝐹y)))
3 19.40 1519 . . . . . . 7 (x((x A x𝐹y) x B x𝐹y)) → (x(x A x𝐹y) xx B x𝐹y)))
42, 3sylbi 114 . . . . . 6 (x((x A ¬ x B) x𝐹y) → (x(x A x𝐹y) xx B x𝐹y)))
5 nfv 1418 . . . . . . . . . . 11 xFun 𝐹
6 nfe1 1382 . . . . . . . . . . 11 xx(x𝐹y ¬ x B)
75, 6nfan 1454 . . . . . . . . . 10 x(Fun 𝐹 x(x𝐹y ¬ x B))
8 funmo 4860 . . . . . . . . . . . . . 14 (Fun 𝐹∃*x y𝐹x)
9 vex 2554 . . . . . . . . . . . . . . . 16 y V
10 vex 2554 . . . . . . . . . . . . . . . 16 x V
119, 10brcnv 4461 . . . . . . . . . . . . . . 15 (y𝐹xx𝐹y)
1211mobii 1934 . . . . . . . . . . . . . 14 (∃*x y𝐹x∃*x x𝐹y)
138, 12sylib 127 . . . . . . . . . . . . 13 (Fun 𝐹∃*x x𝐹y)
14 mopick 1975 . . . . . . . . . . . . 13 ((∃*x x𝐹y x(x𝐹y ¬ x B)) → (x𝐹y → ¬ x B))
1513, 14sylan 267 . . . . . . . . . . . 12 ((Fun 𝐹 x(x𝐹y ¬ x B)) → (x𝐹y → ¬ x B))
1615con2d 554 . . . . . . . . . . 11 ((Fun 𝐹 x(x𝐹y ¬ x B)) → (x B → ¬ x𝐹y))
17 imnan 623 . . . . . . . . . . 11 ((x B → ¬ x𝐹y) ↔ ¬ (x B x𝐹y))
1816, 17sylib 127 . . . . . . . . . 10 ((Fun 𝐹 x(x𝐹y ¬ x B)) → ¬ (x B x𝐹y))
197, 18alrimi 1412 . . . . . . . . 9 ((Fun 𝐹 x(x𝐹y ¬ x B)) → x ¬ (x B x𝐹y))
2019ex 108 . . . . . . . 8 (Fun 𝐹 → (x(x𝐹y ¬ x B) → x ¬ (x B x𝐹y)))
21 exancom 1496 . . . . . . . 8 (x(x𝐹y ¬ x B) ↔ xx B x𝐹y))
22 alnex 1385 . . . . . . . 8 (x ¬ (x B x𝐹y) ↔ ¬ x(x B x𝐹y))
2320, 21, 223imtr3g 193 . . . . . . 7 (Fun 𝐹 → (xx B x𝐹y) → ¬ x(x B x𝐹y)))
2423anim2d 320 . . . . . 6 (Fun 𝐹 → ((x(x A x𝐹y) xx B x𝐹y)) → (x(x A x𝐹y) ¬ x(x B x𝐹y))))
254, 24syl5 28 . . . . 5 (Fun 𝐹 → (x((x A ¬ x B) x𝐹y) → (x(x A x𝐹y) ¬ x(x B x𝐹y))))
26 df-rex 2306 . . . . . 6 (x (AB)x𝐹yx(x (AB) x𝐹y))
27 eldif 2921 . . . . . . . 8 (x (AB) ↔ (x A ¬ x B))
2827anbi1i 431 . . . . . . 7 ((x (AB) x𝐹y) ↔ ((x A ¬ x B) x𝐹y))
2928exbii 1493 . . . . . 6 (x(x (AB) x𝐹y) ↔ x((x A ¬ x B) x𝐹y))
3026, 29bitri 173 . . . . 5 (x (AB)x𝐹yx((x A ¬ x B) x𝐹y))
31 df-rex 2306 . . . . . 6 (x A x𝐹yx(x A x𝐹y))
32 df-rex 2306 . . . . . . 7 (x B x𝐹yx(x B x𝐹y))
3332notbii 593 . . . . . 6 x B x𝐹y ↔ ¬ x(x B x𝐹y))
3431, 33anbi12i 433 . . . . 5 ((x A x𝐹y ¬ x B x𝐹y) ↔ (x(x A x𝐹y) ¬ x(x B x𝐹y)))
3525, 30, 343imtr4g 194 . . . 4 (Fun 𝐹 → (x (AB)x𝐹y → (x A x𝐹y ¬ x B x𝐹y)))
3635ss2abdv 3007 . . 3 (Fun 𝐹 → {yx (AB)x𝐹y} ⊆ {y ∣ (x A x𝐹y ¬ x B x𝐹y)})
37 dfima2 4613 . . 3 (𝐹 “ (AB)) = {yx (AB)x𝐹y}
38 dfima2 4613 . . . . 5 (𝐹A) = {yx A x𝐹y}
39 dfima2 4613 . . . . 5 (𝐹B) = {yx B x𝐹y}
4038, 39difeq12i 3054 . . . 4 ((𝐹A) ∖ (𝐹B)) = ({yx A x𝐹y} ∖ {yx B x𝐹y})
41 difab 3200 . . . 4 ({yx A x𝐹y} ∖ {yx B x𝐹y}) = {y ∣ (x A x𝐹y ¬ x B x𝐹y)}
4240, 41eqtri 2057 . . 3 ((𝐹A) ∖ (𝐹B)) = {y ∣ (x A x𝐹y ¬ x B x𝐹y)}
4336, 37, 423sstr4g 2980 . 2 (Fun 𝐹 → (𝐹 “ (AB)) ⊆ ((𝐹A) ∖ (𝐹B)))
44 imadiflem 4921 . . 3 ((𝐹A) ∖ (𝐹B)) ⊆ (𝐹 “ (AB))
4544a1i 9 . 2 (Fun 𝐹 → ((𝐹A) ∖ (𝐹B)) ⊆ (𝐹 “ (AB)))
4643, 45eqssd 2956 1 (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∖ (𝐹B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  {cab 2023  wrex 2301  cdif 2908  wss 2911   class class class wbr 3755  ccnv 4287  cima 4291  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by:  resdif  5091  difpreima  5237
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