Theorem dmcosseq 3357

Description: Domain of a composition.

Assertion

Ref	Expression
dmcosseq	⊢ (ran B ⊆ dom A → dom ( A ∘ B) = dom B)

Proof of Theorem dmcosseq

Step	Hyp	Ref	Expression
1		hbe1 1014	. . . . . . . 8 ⊢ (∃x xBz → ∀x∃x xBz)
2		ax-17 969	. . . . . . . 8 ⊢ (∃y zAy → ∀x∃y zAy)
3	1, 2	hbim 1005	. . . . . . 7 ⊢ ((∃x xBz → ∃y zAy) → ∀x(∃x xBz → ∃y zAy))
4	3	hbal 1003	. . . . . 6 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀x∀z(∃x xBz → ∃y zAy))
5		hba1 1001	. . . . . . 7 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀z∀z(∃x xBz → ∃y zAy))
6		19.8a 1027	. . . . . . . . . 10 ⊢ (xBz → ∃x xBz)
7	6	imim1i 16	. . . . . . . . 9 ⊢ ((∃x xBz → ∃y zAy) → (xBz → ∃y zAy))
8	7	ancld 298	. . . . . . . 8 ⊢ ((∃x xBz → ∃y zAy) → (xBz → (xBz ⋀ ∃y zAy)))
9	8	a4s 982	. . . . . . 7 ⊢ (∀z(∃x xBz → ∃y zAy) → (xBz → (xBz ⋀ ∃y zAy)))
10	5, 9	19.22d 1060	. . . . . 6 ⊢ (∀z(∃x xBz → ∃y zAy) → (∃z xBz → ∃z(xBz ⋀ ∃y zAy)))
11	4, 10	19.21ai 996	. . . . 5 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀x(∃z xBz → ∃z(xBz ⋀ ∃y zAy)))
12		pm3.26 319	. . . . . . 7 ⊢ ((xBz ⋀ ∃y zAy) → xBz)
13	12	19.22i 1038	. . . . . 6 ⊢ (∃z(xBz ⋀ ∃y zAy) → ∃z xBz)
14	13	ax-gen 961	. . . . 5 ⊢ ∀x(∃z(xBz ⋀ ∃y zAy) → ∃z xBz)
15	11, 14	jctil 292	. . . 4 ⊢ (∀z(∃x xBz → ∃y zAy) → (∀x(∃z(xBz ⋀ ∃y zAy) → ∃z xBz) ⋀ ∀x(∃z xBz → ∃z(xBz ⋀ ∃y zAy))))
16		albi 1105	. . . 4 ⊢ (∀x(∃z(xBz ⋀ ∃y zAy) ↔ ∃z xBz) ↔ (∀x(∃z(xBz ⋀ ∃y zAy) → ∃z xBz) ⋀ ∀x(∃z xBz → ∃z(xBz ⋀ ∃y zAy))))
17	15, 16	sylibr 200	. . 3 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀x(∃z(xBz ⋀ ∃y zAy) ↔ ∃z xBz))
18		visset 1809	. . . . . 6 ⊢ z ∈ V
19	18	elrn 3344	. . . . 5 ⊢ (z ∈ ran B ↔ ∃x xBz)
20	18	eldm 3302	. . . . 5 ⊢ (z ∈ dom A ↔ ∃y zAy)
21	19, 20	imbi12i 188	. . . 4 ⊢ ((z ∈ ran B → z ∈ dom A) ↔ (∃x xBz → ∃y zAy))
22	21	albii 997	. . 3 ⊢ (∀z(z ∈ ran B → z ∈ dom A) ↔ ∀z(∃x xBz → ∃y zAy))
23		visset 1809	. . . . . . 7 ⊢ x ∈ V
24	23	eldm2 3303	. . . . . 6 ⊢ (x ∈ dom ( A ∘ B) ↔ ∃y⟨x, y⟩ ∈ (A ∘ B))
25		visset 1809	. . . . . . . 8 ⊢ y ∈ V
26	23, 25	opelco 3283	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (A ∘ B) ↔ ∃z(xBz ⋀ zAy))
27	26	exbii 1049	. . . . . 6 ⊢ (∃y⟨x, y⟩ ∈ (A ∘ B) ↔ ∃y∃z(xBz ⋀ zAy))
28		excom 1044	. . . . . . 7 ⊢ (∃y∃z(xBz ⋀ zAy) ↔ ∃z∃y(xBz ⋀ zAy))
29		19.42v 1306	. . . . . . . 8 ⊢ (∃y(xBz ⋀ zAy) ↔ (xBz ⋀ ∃y zAy))
30	29	exbii 1049	. . . . . . 7 ⊢ (∃z∃y(xBz ⋀ zAy) ↔ ∃z(xBz ⋀ ∃y zAy))
31	28, 30	bitr 173	. . . . . 6 ⊢ (∃y∃z(xBz ⋀ zAy) ↔ ∃z(xBz ⋀ ∃y zAy))
32	24, 27, 31	3bitr 177	. . . . 5 ⊢ (x ∈ dom ( A ∘ B) ↔ ∃z(xBz ⋀ ∃y zAy))
33	23	eldm 3302	. . . . 5 ⊢ (x ∈ dom B ↔ ∃z xBz)
34	32, 33	bibi12i 609	. . . 4 ⊢ ((x ∈ dom ( A ∘ B) ↔ x ∈ dom B) ↔ (∃z(xBz ⋀ ∃y zAy) ↔ ∃z xBz))
35	34	albii 997	. . 3 ⊢ (∀x(x ∈ dom ( A ∘ B) ↔ x ∈ dom B) ↔ ∀x(∃z(xBz ⋀ ∃y zAy) ↔ ∃z xBz))
36	17, 22, 35	3imtr4 219	. 2 ⊢ (∀z(z ∈ ran B → z ∈ dom A) → ∀x(x ∈ dom ( A ∘ B) ↔ x ∈ dom B))
37		dfss2 2054	. 2 ⊢ (ran B ⊆ dom A ↔ ∀z(z ∈ ran B → z ∈ dom A))
38		dfcleq 1468	. 2 ⊢ (dom ( A ∘ B) = dom B ↔ ∀x(x ∈ dom ( A ∘ B) ↔ x ∈ dom B))
39	36, 37, 38	3imtr4 219	1 ⊢ (ran B ⊆ dom A → dom ( A ∘ B) = dom B)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978   ⊆ wss 2043  ⟨cop 2407   class class class wbr 2614  dom cdm 3165  ran crn 3166   ∘ ccom 3169

This theorem is referenced by:  dmcoeq 3358  fnco 3587  fco 3627  cncfmet1 7858

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184

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