Theorem imaco 6227

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 3055	. . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥))
2		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
3	2	elima 6039	. . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥)
4		vex 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
5	4, 2	brco 5837	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
6	5	rexbii 3077	. . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
7		rexcom4 3265	. . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
8		r19.41v 3168	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
9	8	exbii 1848	. . . . 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 6039	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥)
12		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
13	12	elima 6039	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦)
14	13	anbi1i 624	. . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
15	14	exbii 1848	. . . 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 2727	1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054   class class class wbr 5110   “ cima 5644   ∘ ccom 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  fvco2  6961  suppco  8188  fipreima  9316  fsuppcolem  9359  psgnunilem1  19430  gsumzf1o  19849  dprdf1o  19971  frlmup3  21716  f1lindf  21738  lindfmm  21743  cnco  23160  cnpco  23161  ptrescn  23533  xkoco1cn  23551  xkoco2cn  23552  xkococnlem  23553  qtopcn  23608  fmco  23855  uniioombllem3  25493  cncombf  25566  deg1val  26008  ofpreima  32596  mbfmco  34262  eulerpartlemmf  34373  erdsze2lem2  35198  cvmliftmolem1  35275  cvmlift2lem9a  35297  cvmlift2lem9  35305  mclsppslem  35577  bj-imdirco  37185  poimirlem15  37636  poimirlem16  37637  poimirlem19  37640  cnambfre  37669  ftc1anclem3  37696  aks6d1c6lem4  42168  aks6d1c6lem5  42172  trclimalb2  43722  brtrclfv2  43723  frege97d  43748  frege109d  43753  frege131d  43760  extoimad  44160  imo72b2lem0  44161  imo72b2lem2  44163  imo72b2lem1  44165  imo72b2  44168  limccog  45625  smfco  46807  afv2co2  47262  grimco  47893