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Theorem imadiflem 4921
Description: One direction of imadif 4922. This direction does not require Fun 𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
Assertion
Ref Expression
imadiflem ((𝐹A) ∖ (𝐹B)) ⊆ (𝐹 “ (AB))

Proof of Theorem imadiflem
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2306 . . . 4 (x A x𝐹yx(x A x𝐹y))
2 df-rex 2306 . . . . 5 (x B x𝐹yx(x B x𝐹y))
32notbii 593 . . . 4 x B x𝐹y ↔ ¬ x(x B x𝐹y))
4 alnex 1385 . . . . . . 7 (x ¬ (x B x𝐹y) ↔ ¬ x(x B x𝐹y))
5 19.29r 1509 . . . . . . 7 ((x(x A x𝐹y) x ¬ (x B x𝐹y)) → x((x A x𝐹y) ¬ (x B x𝐹y)))
64, 5sylan2br 272 . . . . . 6 ((x(x A x𝐹y) ¬ x(x B x𝐹y)) → x((x A x𝐹y) ¬ (x B x𝐹y)))
7 simpl 102 . . . . . . . . 9 (((x A x𝐹y) ¬ (x B x𝐹y)) → (x A x𝐹y))
8 simplr 482 . . . . . . . . . 10 (((x A x𝐹y) ¬ (x B x𝐹y)) → x𝐹y)
9 simpr 103 . . . . . . . . . . 11 (((x A x𝐹y) ¬ (x B x𝐹y)) → ¬ (x B x𝐹y))
10 ancom 253 . . . . . . . . . . . . 13 ((x B x𝐹y) ↔ (x𝐹y x B))
1110notbii 593 . . . . . . . . . . . 12 (¬ (x B x𝐹y) ↔ ¬ (x𝐹y x B))
12 imnan 623 . . . . . . . . . . . 12 ((x𝐹y → ¬ x B) ↔ ¬ (x𝐹y x B))
1311, 12bitr4i 176 . . . . . . . . . . 11 (¬ (x B x𝐹y) ↔ (x𝐹y → ¬ x B))
149, 13sylib 127 . . . . . . . . . 10 (((x A x𝐹y) ¬ (x B x𝐹y)) → (x𝐹y → ¬ x B))
158, 14mpd 13 . . . . . . . . 9 (((x A x𝐹y) ¬ (x B x𝐹y)) → ¬ x B)
167, 15, 8jca32 293 . . . . . . . 8 (((x A x𝐹y) ¬ (x B x𝐹y)) → ((x A x𝐹y) x B x𝐹y)))
17 eldif 2921 . . . . . . . . . 10 (x (AB) ↔ (x A ¬ x B))
1817anbi1i 431 . . . . . . . . 9 ((x (AB) x𝐹y) ↔ ((x A ¬ x B) x𝐹y))
19 anandir 525 . . . . . . . . 9 (((x A ¬ x B) x𝐹y) ↔ ((x A x𝐹y) x B x𝐹y)))
2018, 19bitri 173 . . . . . . . 8 ((x (AB) x𝐹y) ↔ ((x A x𝐹y) x B x𝐹y)))
2116, 20sylibr 137 . . . . . . 7 (((x A x𝐹y) ¬ (x B x𝐹y)) → (x (AB) x𝐹y))
2221eximi 1488 . . . . . 6 (x((x A x𝐹y) ¬ (x B x𝐹y)) → x(x (AB) x𝐹y))
236, 22syl 14 . . . . 5 ((x(x A x𝐹y) ¬ x(x B x𝐹y)) → x(x (AB) x𝐹y))
24 df-rex 2306 . . . . 5 (x (AB)x𝐹yx(x (AB) x𝐹y))
2523, 24sylibr 137 . . . 4 ((x(x A x𝐹y) ¬ x(x B x𝐹y)) → x (AB)x𝐹y)
261, 3, 25syl2anb 275 . . 3 ((x A x𝐹y ¬ x B x𝐹y) → x (AB)x𝐹y)
2726ss2abi 3006 . 2 {y ∣ (x A x𝐹y ¬ x B x𝐹y)} ⊆ {yx (AB)x𝐹y}
28 dfima2 4613 . . . 4 (𝐹A) = {yx A x𝐹y}
29 dfima2 4613 . . . 4 (𝐹B) = {yx B x𝐹y}
3028, 29difeq12i 3054 . . 3 ((𝐹A) ∖ (𝐹B)) = ({yx A x𝐹y} ∖ {yx B x𝐹y})
31 difab 3200 . . 3 ({yx A x𝐹y} ∖ {yx B x𝐹y}) = {y ∣ (x A x𝐹y ¬ x B x𝐹y)}
3230, 31eqtri 2057 . 2 ((𝐹A) ∖ (𝐹B)) = {y ∣ (x A x𝐹y ¬ x B x𝐹y)}
33 dfima2 4613 . 2 (𝐹 “ (AB)) = {yx (AB)x𝐹y}
3427, 32, 333sstr4i 2978 1 ((𝐹A) ∖ (𝐹B)) ⊆ (𝐹 “ (AB))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wal 1240  wex 1378   wcel 1390  {cab 2023  wrex 2301  cdif 2908  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-in1 544  ax-in2 545  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-bnd 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-fal 1248  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-rab 2309  df-v 2553  df-dif 2914  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:  imadif  4922
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