Theorem imaco 6282

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	⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))

Proof of Theorem imaco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3077	. . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥))
2		vex 3492	. . . 4 ⊢ 𝑥 ∈ V
3	2	elima 6094	. . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥)
4		vex 3492	. . . . . . 7 ⊢ 𝑧 ∈ V
5	4, 2	brco 5895	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
6	5	rexbii 3100	. . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
7		rexcom4 3294	. . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
8		r19.41v 3195	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
9	8	exbii 1846	. . . . 5 ⊢ (∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
10	6, 7, 9	3bitri 297	. . . 4 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
11	2	elima 6094	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥)
12		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
13	12	elima 6094	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦)
14	13	anbi1i 623	. . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
15	14	exbii 1846	. . . 4 ⊢ (∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
16	10, 11, 15	3bitr4i 303	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥))
17	1, 3, 16	3bitr4ri 304	. 2 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)))
18	17	eqriv 2737	1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166   “ cima 5703   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  fvco2  7019  suppco  8247  fipreima  9428  fsuppcolem  9470  psgnunilem1  19535  gsumzf1o  19954  dprdf1o  20076  frlmup3  21843  f1lindf  21865  lindfmm  21870  cnco  23295  cnpco  23296  ptrescn  23668  xkoco1cn  23686  xkoco2cn  23687  xkococnlem  23688  qtopcn  23743  fmco  23990  uniioombllem3  25639  cncombf  25712  deg1val  26155  ofpreima  32683  mbfmco  34229  eulerpartlemmf  34340  erdsze2lem2  35172  cvmliftmolem1  35249  cvmlift2lem9a  35271  cvmlift2lem9  35279  mclsppslem  35551  bj-imdirco  37156  poimirlem15  37595  poimirlem16  37596  poimirlem19  37599  cnambfre  37628  ftc1anclem3  37655  aks6d1c6lem4  42130  aks6d1c6lem5  42134  trclimalb2  43688  brtrclfv2  43689  frege97d  43714  frege109d  43719  frege131d  43726  extoimad  44126  imo72b2lem0  44127  imo72b2lem2  44129  imo72b2lem1  44131  imo72b2  44134  limccog  45541  smfco  46723  afv2co2  47172  grimco  47764