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Theorem imakexg 4299
 Description: The image of a set under a set is a set. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
imakexg ((A V B W) → (Ak B) V)

Proof of Theorem imakexg
StepHypRef Expression
1 dfimak2 4298 . 2 (Ak B) = ∼ P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V)))
2 1cex 4142 . . . . . 6 1c V
3 vvex 4109 . . . . . 6 V V
42, 3xpkex 4289 . . . . 5 (1c ×k V) V
54complex 4104 . . . 4 ∼ (1c ×k V) V
6 xpkexg 4288 . . . . . . 7 ((B W V V) → (B ×k V) V)
73, 6mpan2 652 . . . . . 6 (B W → (B ×k V) V)
8 inexg 4100 . . . . . 6 ((A V (B ×k V) V) → (A ∩ (B ×k V)) V)
97, 8sylan2 460 . . . . 5 ((A V B W) → (A ∩ (B ×k V)) V)
10 complexg 4099 . . . . 5 ((A ∩ (B ×k V)) V → ∼ (A ∩ (B ×k V)) V)
11 sikexg 4296 . . . . 5 ( ∼ (A ∩ (B ×k V)) V → SIk ∼ (A ∩ (B ×k V)) V)
129, 10, 113syl 18 . . . 4 ((A V B W) → SIk ∼ (A ∩ (B ×k V)) V)
13 unexg 4101 . . . 4 (( ∼ (1c ×k V) V SIk ∼ (A ∩ (B ×k V)) V) → ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V)
145, 12, 13sylancr 644 . . 3 ((A V B W) → ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V)
15 p6exg 4290 . . 3 (( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V → P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V)
16 complexg 4099 . . 3 ( P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V → ∼ P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V)
1714, 15, 163syl 18 . 2 ((A V B W) → ∼ P6 ( ∼ (1c ×k V) ∪ SIk ∼ (A ∩ (B ×k V))) V)
181, 17syl5eqel 2437 1 ((A V B W) → (Ak B) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207   ∩ cin 3208  1cc1c 4134   ×k cxpk 4174   P6 cp6 4178   “k cimak 4179   SIk csik 4181 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-si 4083  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192 This theorem is referenced by:  imakex  4300  pw1exg  4302  cokexg  4309  imagekexg  4311  uniexg  4316  intexg  4319  pwexg  4328  addcexg  4393  phiexg  4571  opexg  4587  proj1exg  4591  proj2exg  4592  imaexg  4746  coexg  4749  siexg  4752
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