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Theorem imain 5212
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 5211 . . 3 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
21a1i 9 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵)))
3 eeanv 1905 . . . . . 6 (∃𝑥𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
4 simprll 527 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐴)
5 simpr 109 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑥𝐹𝑦) → 𝑥𝐹𝑦)
6 simpr 109 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑧𝐹𝑦) → 𝑧𝐹𝑦)
75, 6anim12i 336 . . . . . . . . . . . . 13 (((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥𝐹𝑦𝑧𝐹𝑦))
8 funcnveq 5193 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
98biimpi 119 . . . . . . . . . . . . . . . 16 (Fun 𝐹 → ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
10919.21bi 1538 . . . . . . . . . . . . . . 15 (Fun 𝐹 → ∀𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
111019.21bbi 1539 . . . . . . . . . . . . . 14 (Fun 𝐹 → ((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
1211imp 123 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥𝐹𝑦𝑧𝐹𝑦)) → 𝑥 = 𝑧)
137, 12sylan2 284 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥 = 𝑧)
14 simprrl 529 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑧𝐵)
1513, 14eqeltrd 2217 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐵)
16 elin 3263 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
174, 15, 16sylanbrc 414 . . . . . . . . . 10 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥 ∈ (𝐴𝐵))
18 simprlr 528 . . . . . . . . . 10 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐹𝑦)
1917, 18jca 304 . . . . . . . . 9 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2019ex 114 . . . . . . . 8 (Fun 𝐹 → (((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2120exlimdv 1792 . . . . . . 7 (Fun 𝐹 → (∃𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2221eximdv 1853 . . . . . 6 (Fun 𝐹 → (∃𝑥𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
233, 22syl5bir 152 . . . . 5 (Fun 𝐹 → ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
24 df-rex 2423 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
25 df-rex 2423 . . . . . 6 (∃𝑧𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧𝐵𝑧𝐹𝑦))
2624, 25anbi12i 456 . . . . 5 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
27 df-rex 2423 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2823, 26, 273imtr4g 204 . . . 4 (Fun 𝐹 → ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦))
2928ss2abdv 3174 . . 3 (Fun 𝐹 → {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦})
30 dfima2 4890 . . . . 5 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
31 dfima2 4890 . . . . 5 (𝐹𝐵) = {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}
3230, 31ineq12i 3279 . . . 4 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦})
33 inab 3348 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
3432, 33eqtri 2161 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
35 dfima2 4890 . . 3 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
3629, 34, 353sstr4g 3144 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
372, 36eqssd 3118 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1330   = wceq 1332  wex 1469  wcel 1481  {cab 2126  wrex 2418  cin 3074  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-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-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-v 2691  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:  inpreima  5553
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