Theorem imaco 6106

Description: Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

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 3146	. . 3 ⊢ (∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥 ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥))
2		vex 3499	. . . 4 ⊢ 𝑥 ∈ V
3	2	elima 5936	. . 3 ⊢ (𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)) ↔ ∃𝑦 ∈ (𝐵 “ 𝐶)𝑦𝐴𝑥)
4		rexcom4 3251	. . . . 5 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
5		r19.41v 3349	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
6	5	exbii 1848	. . . . 5 ⊢ (∃𝑦∃𝑧 ∈ 𝐶 (𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
7	4, 6	bitri 277	. . . 4 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
8	2	elima 5936	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥)
9		vex 3499	. . . . . . 7 ⊢ 𝑧 ∈ V
10	9, 2	brco 5743	. . . . . 6 ⊢ (𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
11	10	rexbii 3249	. . . . 5 ⊢ (∃𝑧 ∈ 𝐶 𝑧(𝐴 ∘ 𝐵)𝑥 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
12	8, 11	bitri 277	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 ∃𝑦(𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
13		vex 3499	. . . . . . 7 ⊢ 𝑦 ∈ V
14	13	elima 5936	. . . . . 6 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧 ∈ 𝐶 𝑧𝐵𝑦)
15	14	anbi1i 625	. . . . 5 ⊢ ((𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ (∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
16	15	exbii 1848	. . . 4 ⊢ (∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥) ↔ ∃𝑦(∃𝑧 ∈ 𝐶 𝑧𝐵𝑦 ∧ 𝑦𝐴𝑥))
17	7, 12, 16	3bitr4i 305	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ ∃𝑦(𝑦 ∈ (𝐵 “ 𝐶) ∧ 𝑦𝐴𝑥))
18	1, 3, 17	3bitr4ri 306	. 2 ⊢ (𝑥 ∈ ((𝐴 ∘ 𝐵) “ 𝐶) ↔ 𝑥 ∈ (𝐴 “ (𝐵 “ 𝐶)))
19	18	eqriv 2820	1 ⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃wrex 3141   class class class wbr 5068   “ cima 5560   ∘ ccom 5561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570

This theorem is referenced by:  fvco2  6760  suppco  7872  supp0cosupp0OLD  7875  imacosuppOLD  7877  fipreima  8832  fsuppcolem  8866  psgnunilem1  18623  gsumzf1o  19034  dprdf1o  19156  frlmup3  20946  f1lindf  20968  lindfmm  20973  cnco  21876  cnpco  21877  ptrescn  22249  xkoco1cn  22267  xkoco2cn  22268  xkococnlem  22269  qtopcn  22324  fmco  22571  uniioombllem3  24188  cncombf  24261  deg1val  24692  ofpreima  30412  mbfmco  31524  eulerpartlemmf  31635  erdsze2lem2  32453  cvmliftmolem1  32530  cvmlift2lem9a  32552  cvmlift2lem9  32560  mclsppslem  32832  poimirlem15  34909  poimirlem16  34910  poimirlem19  34913  cnambfre  34942  ftc1anclem3  34971  trclimalb2  40078  brtrclfv2  40079  frege97d  40104  frege109d  40109  frege131d  40116  extoimad  40522  imo72b2lem0  40523  imo72b2lem2  40525  imo72b2lem1  40528  imo72b2  40532  limccog  41908  smfco  43084  afv2co2  43463  isomgrtrlem  44010