MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imaco Structured version   Visualization version   GIF version

Theorem imaco 6270
Description: Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) (Proof shortened by Wolf Lammen, 16-May-2025.)
Assertion
Ref Expression
imaco ((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))

Proof of Theorem imaco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 3070 . . 3 (∃𝑦 ∈ (𝐵𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵𝐶) ∧ 𝑦𝐴𝑥))
2 vex 3483 . . . 4 𝑥 ∈ V
32elima 6082 . . 3 (𝑥 ∈ (𝐴 “ (𝐵𝐶)) ↔ ∃𝑦 ∈ (𝐵𝐶)𝑦𝐴𝑥)
4 vex 3483 . . . . . . 7 𝑧 ∈ V
54, 2brco 5880 . . . . . 6 (𝑧(𝐴𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦𝑦𝐴𝑥))
65rexbii 3093 . . . . 5 (∃𝑧𝐶 𝑧(𝐴𝐵)𝑥 ↔ ∃𝑧𝐶𝑦(𝑧𝐵𝑦𝑦𝐴𝑥))
7 rexcom4 3287 . . . . 5 (∃𝑧𝐶𝑦(𝑧𝐵𝑦𝑦𝐴𝑥) ↔ ∃𝑦𝑧𝐶 (𝑧𝐵𝑦𝑦𝐴𝑥))
8 r19.41v 3188 . . . . . 6 (∃𝑧𝐶 (𝑧𝐵𝑦𝑦𝐴𝑥) ↔ (∃𝑧𝐶 𝑧𝐵𝑦𝑦𝐴𝑥))
98exbii 1847 . . . . 5 (∃𝑦𝑧𝐶 (𝑧𝐵𝑦𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧𝐶 𝑧𝐵𝑦𝑦𝐴𝑥))
106, 7, 93bitri 297 . . . 4 (∃𝑧𝐶 𝑧(𝐴𝐵)𝑥 ↔ ∃𝑦(∃𝑧𝐶 𝑧𝐵𝑦𝑦𝐴𝑥))
112elima 6082 . . . 4 (𝑥 ∈ ((𝐴𝐵) “ 𝐶) ↔ ∃𝑧𝐶 𝑧(𝐴𝐵)𝑥)
12 vex 3483 . . . . . . 7 𝑦 ∈ V
1312elima 6082 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ ∃𝑧𝐶 𝑧𝐵𝑦)
1413anbi1i 624 . . . . 5 ((𝑦 ∈ (𝐵𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧𝐶 𝑧𝐵𝑦𝑦𝐴𝑥))
1514exbii 1847 . . . 4 (∃𝑦(𝑦 ∈ (𝐵𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧𝐶 𝑧𝐵𝑦𝑦𝐴𝑥))
1610, 11, 153bitr4i 303 . . 3 (𝑥 ∈ ((𝐴𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵𝐶) ∧ 𝑦𝐴𝑥))
171, 3, 163bitr4ri 304 . 2 (𝑥 ∈ ((𝐴𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵𝐶)))
1817eqriv 2733 1 ((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wex 1778  wcel 2107  wrex 3069   class class class wbr 5142  cima 5687  ccom 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  fvco2  7005  suppco  8232  fipreima  9399  fsuppcolem  9442  psgnunilem1  19512  gsumzf1o  19931  dprdf1o  20053  frlmup3  21821  f1lindf  21843  lindfmm  21848  cnco  23275  cnpco  23276  ptrescn  23648  xkoco1cn  23666  xkoco2cn  23667  xkococnlem  23668  qtopcn  23723  fmco  23970  uniioombllem3  25621  cncombf  25694  deg1val  26136  ofpreima  32676  mbfmco  34267  eulerpartlemmf  34378  erdsze2lem2  35210  cvmliftmolem1  35287  cvmlift2lem9a  35309  cvmlift2lem9  35317  mclsppslem  35589  bj-imdirco  37192  poimirlem15  37643  poimirlem16  37644  poimirlem19  37647  cnambfre  37676  ftc1anclem3  37703  aks6d1c6lem4  42175  aks6d1c6lem5  42179  trclimalb2  43744  brtrclfv2  43745  frege97d  43770  frege109d  43775  frege131d  43782  extoimad  44182  imo72b2lem0  44183  imo72b2lem2  44185  imo72b2lem1  44187  imo72b2  44190  limccog  45640  smfco  46822  afv2co2  47274  grimco  47885
  Copyright terms: Public domain W3C validator