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

Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anandir 802 . . . . . . 7 (((x A ¬ x B) xFy) ↔ ((x A xFy) x B xFy)))
21exbii 1582 . . . . . 6 (x((x A ¬ x B) xFy) ↔ x((x A xFy) x B xFy)))
3 19.40 1609 . . . . . 6 (x((x A xFy) x B xFy)) → (x(x A xFy) xx B xFy)))
42, 3sylbi 187 . . . . 5 (x((x A ¬ x B) xFy) → (x(x A xFy) xx B xFy)))
5 nfv 1619 . . . . . . . . . 10 xFun F
6 nfe1 1732 . . . . . . . . . 10 xx(xFy ¬ x B)
75, 6nfan 1824 . . . . . . . . 9 x(Fun F x(xFy ¬ x B))
8 funmo 5125 . . . . . . . . . . . . 13 (Fun F∃*x yFx)
9 brcnv 4892 . . . . . . . . . . . . . 14 (yFxxFy)
109mobii 2240 . . . . . . . . . . . . 13 (∃*x yFx∃*x xFy)
118, 10sylib 188 . . . . . . . . . . . 12 (Fun F∃*x xFy)
12 mopick 2266 . . . . . . . . . . . 12 ((∃*x xFy x(xFy ¬ x B)) → (xFy → ¬ x B))
1311, 12sylan 457 . . . . . . . . . . 11 ((Fun F x(xFy ¬ x B)) → (xFy → ¬ x B))
1413con2d 107 . . . . . . . . . 10 ((Fun F x(xFy ¬ x B)) → (x B → ¬ xFy))
15 imnan 411 . . . . . . . . . 10 ((x B → ¬ xFy) ↔ ¬ (x B xFy))
1614, 15sylib 188 . . . . . . . . 9 ((Fun F x(xFy ¬ x B)) → ¬ (x B xFy))
177, 16alrimi 1765 . . . . . . . 8 ((Fun F x(xFy ¬ x B)) → x ¬ (x B xFy))
1817ex 423 . . . . . . 7 (Fun F → (x(xFy ¬ x B) → x ¬ (x B xFy)))
19 exancom 1586 . . . . . . 7 (x(xFy ¬ x B) ↔ xx B xFy))
20 alnex 1543 . . . . . . 7 (x ¬ (x B xFy) ↔ ¬ x(x B xFy))
2118, 19, 203imtr3g 260 . . . . . 6 (Fun F → (xx B xFy) → ¬ x(x B xFy)))
2221anim2d 548 . . . . 5 (Fun F → ((x(x A xFy) xx B xFy)) → (x(x A xFy) ¬ x(x B xFy))))
234, 22syl5 28 . . . 4 (Fun F → (x((x A ¬ x B) xFy) → (x(x A xFy) ¬ x(x B xFy))))
24 19.29r 1597 . . . . . 6 ((x(x A xFy) x ¬ (x B xFy)) → x((x A xFy) ¬ (x B xFy)))
2520, 24sylan2br 462 . . . . 5 ((x(x A xFy) ¬ x(x B xFy)) → x((x A xFy) ¬ (x B xFy)))
26 andi 837 . . . . . . 7 (((x A xFy) x B ¬ xFy)) ↔ (((x A xFy) ¬ x B) ((x A xFy) ¬ xFy)))
27 ianor 474 . . . . . . . 8 (¬ (x B xFy) ↔ (¬ x B ¬ xFy))
2827anbi2i 675 . . . . . . 7 (((x A xFy) ¬ (x B xFy)) ↔ ((x A xFy) x B ¬ xFy)))
29 an32 773 . . . . . . . 8 (((x A ¬ x B) xFy) ↔ ((x A xFy) ¬ x B))
30 pm3.24 852 . . . . . . . . . . 11 ¬ (xFy ¬ xFy)
3130intnan 880 . . . . . . . . . 10 ¬ (x A (xFy ¬ xFy))
32 anass 630 . . . . . . . . . 10 (((x A xFy) ¬ xFy) ↔ (x A (xFy ¬ xFy)))
3331, 32mtbir 290 . . . . . . . . 9 ¬ ((x A xFy) ¬ xFy)
3433biorfi 396 . . . . . . . 8 (((x A xFy) ¬ x B) ↔ (((x A xFy) ¬ x B) ((x A xFy) ¬ xFy)))
3529, 34bitri 240 . . . . . . 7 (((x A ¬ x B) xFy) ↔ (((x A xFy) ¬ x B) ((x A xFy) ¬ xFy)))
3626, 28, 353bitr4i 268 . . . . . 6 (((x A xFy) ¬ (x B xFy)) ↔ ((x A ¬ x B) xFy))
3736exbii 1582 . . . . 5 (x((x A xFy) ¬ (x B xFy)) ↔ x((x A ¬ x B) xFy))
3825, 37sylib 188 . . . 4 ((x(x A xFy) ¬ x(x B xFy)) → x((x A ¬ x B) xFy))
3923, 38impbid1 194 . . 3 (Fun F → (x((x A ¬ x B) xFy) ↔ (x(x A xFy) ¬ x(x B xFy))))
40 elima2 4755 . . . 4 (y (F “ (A B)) ↔ x(x (A B) xFy))
41 eldif 3221 . . . . . 6 (x (A B) ↔ (x A ¬ x B))
4241anbi1i 676 . . . . 5 ((x (A B) xFy) ↔ ((x A ¬ x B) xFy))
4342exbii 1582 . . . 4 (x(x (A B) xFy) ↔ x((x A ¬ x B) xFy))
4440, 43bitri 240 . . 3 (y (F “ (A B)) ↔ x((x A ¬ x B) xFy))
45 eldif 3221 . . . 4 (y ((FA) (FB)) ↔ (y (FA) ¬ y (FB)))
46 elima2 4755 . . . . 5 (y (FA) ↔ x(x A xFy))
47 elima2 4755 . . . . . 6 (y (FB) ↔ x(x B xFy))
4847notbii 287 . . . . 5 y (FB) ↔ ¬ x(x B xFy))
4946, 48anbi12i 678 . . . 4 ((y (FA) ¬ y (FB)) ↔ (x(x A xFy) ¬ x(x B xFy)))
5045, 49bitri 240 . . 3 (y ((FA) (FB)) ↔ (x(x A xFy) ¬ x(x B xFy)))
5139, 44, 503bitr4g 279 . 2 (Fun F → (y (F “ (A B)) ↔ y ((FA) (FB))))
5251eqrdv 2351 1 (Fun F → (F “ (A B)) = ((FA) (FB)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205   ∖ cdif 3206   class class class wbr 4639   “ cima 4722  ◡ccnv 4771  Fun wfun 4775 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-fun 4789 This theorem is referenced by:  imain  5172  resdif  5306
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