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Theorem imain 5082
Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
imain (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))

Proof of Theorem imain
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imainlem 5081 . . 3 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
21a1i 9 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵)))
3 eeanv 1855 . . . . . 6 (∃𝑥𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
4 simprll 504 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐴)
5 simpr 108 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑥𝐹𝑦) → 𝑥𝐹𝑦)
6 simpr 108 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑧𝐹𝑦) → 𝑧𝐹𝑦)
75, 6anim12i 331 . . . . . . . . . . . . 13 (((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥𝐹𝑦𝑧𝐹𝑦))
8 funcnveq 5063 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
98biimpi 118 . . . . . . . . . . . . . . . 16 (Fun 𝐹 → ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
10919.21bi 1495 . . . . . . . . . . . . . . 15 (Fun 𝐹 → ∀𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
111019.21bbi 1496 . . . . . . . . . . . . . 14 (Fun 𝐹 → ((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
1211imp 122 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥𝐹𝑦𝑧𝐹𝑦)) → 𝑥 = 𝑧)
137, 12sylan2 280 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥 = 𝑧)
14 simprrl 506 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑧𝐵)
1513, 14eqeltrd 2164 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐵)
16 elin 3181 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
174, 15, 16sylanbrc 408 . . . . . . . . . 10 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥 ∈ (𝐴𝐵))
18 simprlr 505 . . . . . . . . . 10 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐹𝑦)
1917, 18jca 300 . . . . . . . . 9 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2019ex 113 . . . . . . . 8 (Fun 𝐹 → (((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2120exlimdv 1747 . . . . . . 7 (Fun 𝐹 → (∃𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2221eximdv 1808 . . . . . 6 (Fun 𝐹 → (∃𝑥𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
233, 22syl5bir 151 . . . . 5 (Fun 𝐹 → ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
24 df-rex 2365 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
25 df-rex 2365 . . . . . 6 (∃𝑧𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧𝐵𝑧𝐹𝑦))
2624, 25anbi12i 448 . . . . 5 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
27 df-rex 2365 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2823, 26, 273imtr4g 203 . . . 4 (Fun 𝐹 → ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦))
2928ss2abdv 3092 . . 3 (Fun 𝐹 → {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦})
30 dfima2 4763 . . . . 5 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
31 dfima2 4763 . . . . 5 (𝐹𝐵) = {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}
3230, 31ineq12i 3197 . . . 4 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦})
33 inab 3265 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
3432, 33eqtri 2108 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
35 dfima2 4763 . . 3 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
3629, 34, 353sstr4g 3065 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
372, 36eqssd 3040 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426  wcel 1438  {cab 2074  wrex 2360  cin 2996  wss 2997   class class class wbr 3837  ccnv 4427  cima 4431  Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fun 5004
This theorem is referenced by:  inpreima  5409
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