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Theorem imain 5006
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 5005 . . 3 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
21a1i 9 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵)))
3 eeanv 1821 . . . . . 6 (∃𝑥𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
4 simprll 497 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐴)
5 simpr 107 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑥𝐹𝑦) → 𝑥𝐹𝑦)
6 simpr 107 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑧𝐹𝑦) → 𝑧𝐹𝑦)
75, 6anim12i 325 . . . . . . . . . . . . 13 (((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥𝐹𝑦𝑧𝐹𝑦))
8 funcnveq 4987 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
98biimpi 117 . . . . . . . . . . . . . . . 16 (Fun 𝐹 → ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
10919.21bi 1464 . . . . . . . . . . . . . . 15 (Fun 𝐹 → ∀𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
111019.21bbi 1465 . . . . . . . . . . . . . 14 (Fun 𝐹 → ((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
1211imp 119 . . . . . . . . . . . . 13 ((Fun 𝐹 ∧ (𝑥𝐹𝑦𝑧𝐹𝑦)) → 𝑥 = 𝑧)
137, 12sylan2 274 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥 = 𝑧)
14 simprrl 499 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑧𝐵)
1513, 14eqeltrd 2128 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐵)
16 elin 3151 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
174, 15, 16sylanbrc 402 . . . . . . . . . 10 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥 ∈ (𝐴𝐵))
18 simprlr 498 . . . . . . . . . 10 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐹𝑦)
1917, 18jca 294 . . . . . . . . 9 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2019ex 112 . . . . . . . 8 (Fun 𝐹 → (((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2120exlimdv 1714 . . . . . . 7 (Fun 𝐹 → (∃𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2221eximdv 1774 . . . . . 6 (Fun 𝐹 → (∃𝑥𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
233, 22syl5bir 146 . . . . 5 (Fun 𝐹 → ((∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
24 df-rex 2327 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
25 df-rex 2327 . . . . . 6 (∃𝑧𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧𝐵𝑧𝐹𝑦))
2624, 25anbi12i 441 . . . . 5 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
27 df-rex 2327 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2823, 26, 273imtr4g 198 . . . 4 (Fun 𝐹 → ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦))
2928ss2abdv 3038 . . 3 (Fun 𝐹 → {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦})
30 dfima2 4695 . . . . 5 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
31 dfima2 4695 . . . . 5 (𝐹𝐵) = {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}
3230, 31ineq12i 3161 . . . 4 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦})
33 inab 3230 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
3432, 33eqtri 2074 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
35 dfima2 4695 . . 3 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
3629, 34, 353sstr4g 3011 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
372, 36eqssd 2987 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1255   = wceq 1257  wex 1395  wcel 1407  {cab 2040  wrex 2322  cin 2941  wss 2942   class class class wbr 3789  ccnv 4369  cima 4373  Fun wfun 4921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-fun 4929
This theorem is referenced by:  inpreima  5318
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