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Theorem imainlem 4923
Description: One direction of imain 4924. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
Assertion
Ref Expression
imainlem (𝐹 “ (AB)) ⊆ ((𝐹A) ∩ (𝐹B))

Proof of Theorem imainlem
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2306 . . . . 5 (x (AB)x𝐹yx(x (AB) x𝐹y))
2 elin 3120 . . . . . . . . 9 (x (AB) ↔ (x A x B))
32anbi1i 431 . . . . . . . 8 ((x (AB) x𝐹y) ↔ ((x A x B) x𝐹y))
4 anandir 525 . . . . . . . 8 (((x A x B) x𝐹y) ↔ ((x A x𝐹y) (x B x𝐹y)))
53, 4bitri 173 . . . . . . 7 ((x (AB) x𝐹y) ↔ ((x A x𝐹y) (x B x𝐹y)))
65exbii 1493 . . . . . 6 (x(x (AB) x𝐹y) ↔ x((x A x𝐹y) (x B x𝐹y)))
7 19.40 1519 . . . . . 6 (x((x A x𝐹y) (x B x𝐹y)) → (x(x A x𝐹y) x(x B x𝐹y)))
86, 7sylbi 114 . . . . 5 (x(x (AB) x𝐹y) → (x(x A x𝐹y) x(x B x𝐹y)))
91, 8sylbi 114 . . . 4 (x (AB)x𝐹y → (x(x A x𝐹y) x(x B x𝐹y)))
10 df-rex 2306 . . . . 5 (x A x𝐹yx(x A x𝐹y))
11 df-rex 2306 . . . . 5 (x B x𝐹yx(x B x𝐹y))
1210, 11anbi12i 433 . . . 4 ((x A x𝐹y x B x𝐹y) ↔ (x(x A x𝐹y) x(x B x𝐹y)))
139, 12sylibr 137 . . 3 (x (AB)x𝐹y → (x A x𝐹y x B x𝐹y))
1413ss2abi 3006 . 2 {yx (AB)x𝐹y} ⊆ {y ∣ (x A x𝐹y x B x𝐹y)}
15 dfima2 4613 . 2 (𝐹 “ (AB)) = {yx (AB)x𝐹y}
16 dfima2 4613 . . . 4 (𝐹A) = {yx A x𝐹y}
17 dfima2 4613 . . . 4 (𝐹B) = {yx B x𝐹y}
1816, 17ineq12i 3130 . . 3 ((𝐹A) ∩ (𝐹B)) = ({yx A x𝐹y} ∩ {yx B x𝐹y})
19 inab 3199 . . 3 ({yx A x𝐹y} ∩ {yx B x𝐹y}) = {y ∣ (x A x𝐹y x B x𝐹y)}
2018, 19eqtri 2057 . 2 ((𝐹A) ∩ (𝐹B)) = {y ∣ (x A x𝐹y x B x𝐹y)}
2114, 15, 203sstr4i 2978 1 (𝐹 “ (AB)) ⊆ ((𝐹A) ∩ (𝐹B))
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1378   wcel 1390  {cab 2023  wrex 2301  cin 2910  wss 2911   class class class wbr 3755  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  imain  4924
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