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Theorem imadif 5159
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 563 . . . . . . . 8 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
21exbii 1565 . . . . . . 7 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)))
3 19.40 1591 . . . . . . 7 (∃𝑥((𝑥𝐴𝑥𝐹𝑦) ∧ (¬ 𝑥𝐵𝑥𝐹𝑦)) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
42, 3sylbi 120 . . . . . 6 (∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦) → (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑥𝑥𝐵𝑥𝐹𝑦)))
5 nfv 1489 . . . . . . . . . . 11 𝑥Fun 𝐹
6 nfe1 1453 . . . . . . . . . . 11 𝑥𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)
75, 6nfan 1525 . . . . . . . . . 10 𝑥(Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵))
8 funmo 5094 . . . . . . . . . . . . . 14 (Fun 𝐹 → ∃*𝑥 𝑦𝐹𝑥)
9 vex 2658 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
10 vex 2658 . . . . . . . . . . . . . . . 16 𝑥 ∈ V
119, 10brcnv 4680 . . . . . . . . . . . . . . 15 (𝑦𝐹𝑥𝑥𝐹𝑦)
1211mobii 2010 . . . . . . . . . . . . . 14 (∃*𝑥 𝑦𝐹𝑥 ↔ ∃*𝑥 𝑥𝐹𝑦)
138, 12sylib 121 . . . . . . . . . . . . 13 (Fun 𝐹 → ∃*𝑥 𝑥𝐹𝑦)
14 mopick 2051 . . . . . . . . . . . . 13 ((∃*𝑥 𝑥𝐹𝑦 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1513, 14sylan 279 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐹𝑦 → ¬ 𝑥𝐵))
1615con2d 596 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → (𝑥𝐵 → ¬ 𝑥𝐹𝑦))
17 imnan 662 . . . . . . . . . . 11 ((𝑥𝐵 → ¬ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐵𝑥𝐹𝑦))
1816, 17sylib 121 . . . . . . . . . 10 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ¬ (𝑥𝐵𝑥𝐹𝑦))
197, 18alrimi 1483 . . . . . . . . 9 ((Fun 𝐹 ∧ ∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵)) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦))
2019ex 114 . . . . . . . 8 (Fun 𝐹 → (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) → ∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦)))
21 exancom 1568 . . . . . . . 8 (∃𝑥(𝑥𝐹𝑦 ∧ ¬ 𝑥𝐵) ↔ ∃𝑥𝑥𝐵𝑥𝐹𝑦))
22 alnex 1456 . . . . . . . 8 (∀𝑥 ¬ (𝑥𝐵𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
2320, 21, 223imtr3g 203 . . . . . . 7 (Fun 𝐹 → (∃𝑥𝑥𝐵𝑥𝐹𝑦) → ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
2423anim2d 333 . . . . . 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 2394 . . . . . 6 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
27 eldif 3044 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2827anbi1i 451 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
2928exbii 1565 . . . . . 6 (∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦) ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
3026, 29bitri 183 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐹𝑦))
31 df-rex 2394 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
32 df-rex 2394 . . . . . . 7 (∃𝑥𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
3332notbii 640 . . . . . 6 (¬ ∃𝑥𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦))
3431, 33anbi12i 453 . . . . 5 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥𝐵𝑥𝐹𝑦)))
3525, 30, 343imtr4g 204 . . . 4 (Fun 𝐹 → (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 → (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)))
3635ss2abdv 3134 . . 3 (Fun 𝐹 → {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦} ⊆ {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)})
37 dfima2 4839 . . 3 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
38 dfima2 4839 . . . . 5 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
39 dfima2 4839 . . . . 5 (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}
4038, 39difeq12i 3156 . . . 4 ((𝐹𝐴) ∖ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦})
41 difab 3309 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)}
4240, 41eqtri 2133 . . 3 ((𝐹𝐴) ∖ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥𝐵 𝑥𝐹𝑦)}
4336, 37, 423sstr4g 3104 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∖ (𝐹𝐵)))
44 imadiflem 5158 . . 3 ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵))
4544a1i 9 . 2 (Fun 𝐹 → ((𝐹𝐴) ∖ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
4643, 45eqssd 3078 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1310   = wceq 1312  wex 1449  wcel 1461  ∃*wmo 1974  {cab 2099  wrex 2389  cdif 3032  wss 3035   class class class wbr 3893  ccnv 4496  cima 4500  Fun wfun 5073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-fun 5081
This theorem is referenced by:  resdif  5343  difpreima  5499  phplem4  6700  phplem4dom  6707  phplem4on  6712  cnclima  12228
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