Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  imadif GIF version

 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)))

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anandir 581 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
21exbii 1585 . . . . . . 7 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
3 19.40 1611 . . . . . . 7 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
42, 3sylbi 120 . . . . . 6 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
5 nfv 1509 . . . . . . . . . . 11 𝑥Fun 𝐹
6 nfe1 1473 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)
75, 6nfan 1545 . . . . . . . . . 10 𝑥(Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵))
8 funmo 5145 . . . . . . . . . . . . . 14 (Fun 𝐹 → ∃*𝑥 𝑦𝐹𝑥)
9 vex 2692 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 2692 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
119, 10brcnv 4729 . . . . . . . . . . . . . . 15 (𝑦𝐹𝑥𝑥𝐹𝑦)
1211mobii 2037 . . . . . . . . . . . . . 14 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
138, 12sylib 121 . . . . . . . . . . . . 13 (Fun 𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14 mopick 2078 . . . . . . . . . . . . 13 ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1513, 14sylan 281 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1615con2d 614 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐵 → ¬ 𝑥𝐹𝑦))
17 imnan 680 . . . . . . . . . . 11 ((𝑥𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐵𝑥𝐹𝑦))
1816, 17sylib 121 . . . . . . . . . 10 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
197, 18alrimi 1503 . . . . . . . . 9 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦))
2019ex 114 . . . . . . . 8 (Fun 𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)))
21 exancom 1588 . . . . . . . 8 (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) ↔ ∃𝑥𝑥𝐵𝑥𝐹𝑦))
22 alnex 1476 . . . . . . . 8 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
2320, 21, 223imtr3g 203 . . . . . . 7 (Fun 𝐹 → (∃𝑥𝑥𝐵𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
2423anim2d 335 . . . . . 6 (Fun 𝐹 → ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
254, 24syl5 32 . . . . 5 (Fun 𝐹 → (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))))
26 df-rex 2423 . . . . . 6 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
27 eldif 3084 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2827anbi1i 454 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
2928exbii 1585 . . . . . 6 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
3026, 29bitri 183 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
31 df-rex 2423 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
32 df-rex 2423 . . . . . . 7 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
3332notbii 658 . . . . . 6 (¬ ∃𝑥𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
3431, 33anbi12i 456 . . . . 5 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
3525, 30, 343imtr4g 204 . . . 4 (Fun 𝐹 → (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)))
3635ss2abdv 3174 . . 3 (Fun 𝐹 → {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)})
37 dfima2 4890 . . 3 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
38 dfima2 4890 . . . . 5 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
39 dfima2 4890 . . . . 5 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
4038, 39difeq12i 3196 . . . 4 ((𝐹𝐴) ∖ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
41 difab 3349 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)}
4240, 41eqtri 2161 . . 3 ((𝐹𝐴) ∖ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)}
4336, 37, 423sstr4g 3144 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∖ (𝐹𝐵)))
44 imadiflem 5209 . . 3 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
4544a1i 9 . 2 (Fun 𝐹 → ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
4643, 45eqssd 3118 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃*wmo 2001  {cab 2126  ∃wrex 2418   ∖ cdif 3072   ⊆ wss 3075   class class class wbr 3936  ◡ccnv 4545   “ cima 4549  Fun wfun 5124 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-fun 5132 This theorem is referenced by:  resdif  5396  difpreima  5554  phplem4  6756  phplem4dom  6763  phplem4on  6768  cnclima  12429
 Copyright terms: Public domain W3C validator