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Theorem imain 5032
 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 5031 . . 3 (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵))
21a1i 9 . 2 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) ⊆ ((𝐹𝐴) ∩ (𝐹𝐵)))
3 eeanv 1850 . . . . . 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 5013 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 ↔ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
98biimpi 118 . . . . . . . . . . . . . . . 16 (Fun 𝐹 → ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
10919.21bi 1491 . . . . . . . . . . . . . . 15 (Fun 𝐹 → ∀𝑦𝑧((𝑥𝐹𝑦𝑧𝐹𝑦) → 𝑥 = 𝑧))
111019.21bbi 1492 . . . . . . . . . . . . . 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 2159 . . . . . . . . . . 11 ((Fun 𝐹 ∧ ((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦))) → 𝑥𝐵)
16 elin 3165 . . . . . . . . . . 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 1742 . . . . . . 7 (Fun 𝐹 → (∃𝑧((𝑥𝐴𝑥𝐹𝑦) ∧ (𝑧𝐵𝑧𝐹𝑦)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦)))
2221eximdv 1803 . . . . . 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 2359 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
25 df-rex 2359 . . . . . 6 (∃𝑧𝐵 𝑧𝐹𝑦 ↔ ∃𝑧(𝑧𝐵𝑧𝐹𝑦))
2624, 25anbi12i 448 . . . . 5 ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) ↔ (∃𝑥(𝑥𝐴𝑥𝐹𝑦) ∧ ∃𝑧(𝑧𝐵𝑧𝐹𝑦)))
27 df-rex 2359 . . . . 5 (∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐹𝑦))
2823, 26, 273imtr4g 203 . . . 4 (Fun 𝐹 → ((∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦) → ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦))
2928ss2abdv 3076 . . 3 (Fun 𝐹 → {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦})
30 dfima2 4720 . . . . 5 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
31 dfima2 4720 . . . . 5 (𝐹𝐵) = {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}
3230, 31ineq12i 3181 . . . 4 ((𝐹𝐴) ∩ (𝐹𝐵)) = ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦})
33 inab 3248 . . . 4 ({𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦} ∩ {𝑦 ∣ ∃𝑧𝐵 𝑧𝐹𝑦}) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
3432, 33eqtri 2103 . . 3 ((𝐹𝐴) ∩ (𝐹𝐵)) = {𝑦 ∣ (∃𝑥𝐴 𝑥𝐹𝑦 ∧ ∃𝑧𝐵 𝑧𝐹𝑦)}
35 dfima2 4720 . . 3 (𝐹 “ (𝐴𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴𝐵)𝑥𝐹𝑦}
3629, 34, 353sstr4g 3049 . 2 (Fun 𝐹 → ((𝐹𝐴) ∩ (𝐹𝐵)) ⊆ (𝐹 “ (𝐴𝐵)))
372, 36eqssd 3025 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1283   = wceq 1285  ∃wex 1422   ∈ wcel 1434  {cab 2069  ∃wrex 2354   ∩ cin 2981   ⊆ wss 2982   class class class wbr 3805  ◡ccnv 4390   “ cima 4394  Fun wfun 4946 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-fun 4954 This theorem is referenced by:  inpreima  5345
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