Theorem dmcosseq 2572

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 709	. . . . . . . 8 ⊢ (∃x xBz → ∀x∃x xBz)
2		ax-17 925	. . . . . . . 8 ⊢ (∃y zAy → ∀x∃y zAy)
3	1, 2	hbim 702	. . . . . . 7 ⊢ ((∃x xBz → ∃y zAy) → ∀x(∃x xBz → ∃y zAy))
4	3	hbal 700	. . . . . 6 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀x∀z(∃x xBz → ∃y zAy))
5		hba1 698	. . . . . . 7 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀z∀z(∃x xBz → ∃y zAy))
6		19.8a 712	. . . . . . . . . 10 ⊢ (xBz → ∃x xBz)
7	6	syl4 19	. . . . . . . . 9 ⊢ ((∃x xBz → ∃y zAy) → (xBz → ∃y zAy))
8	7	ancld 246	. . . . . . . 8 ⊢ ((∃x xBz → ∃y zAy) → (xBz → (xBz ∧ ∃y zAy)))
9	8	a4s 682	. . . . . . 7 ⊢ (∀z(∃x xBz → ∃y zAy) → (xBz → (xBz ∧ ∃y zAy)))
10	5, 9	19.22d 744	. . . . . 6 ⊢ (∀z(∃x xBz → ∃y zAy) → (∃z xBz → ∃z(xBz ∧ ∃y zAy)))
11	4, 10	19.21ai 740	. . . . 5 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀x(∃z xBz → ∃z(xBz ∧ ∃y zAy)))
12		pm3.26 256	. . . . . . 7 ⊢ ((xBz ∧ ∃y zAy) → xBz)
13	12	19.22i 723	. . . . . 6 ⊢ (∃z(xBz ∧ ∃y zAy) → ∃z xBz)
14	13	ax-gen 677	. . . . 5 ⊢ ∀x(∃z(xBz ∧ ∃y zAy) → ∃z xBz)
15	11, 14	jctil 240	. . . 4 ⊢ (∀z(∃x xBz → ∃y zAy) → (∀x(∃z(xBz ∧ ∃y zAy) → ∃z xBz) ∧ ∀x(∃z xBz → ∃z(xBz ∧ ∃y zAy))))
16		albi 785	. . . 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 175	. . 3 ⊢ (∀z(∃x xBz → ∃y zAy) → ∀x(∃z(xBz ∧ ∃y zAy) ↔ ∃z xBz))
18		visset 1350	. . . . . 6 ⊢ z ∈ V
19	18	elrn2 2563	. . . . 5 ⊢ (z ∈ ran B ↔ ∃x xBz)
20	18	eldm 2527	. . . . 5 ⊢ (z ∈ dom A ↔ ∃y zAy)
21	19, 20	imbi12i 163	. . . 4 ⊢ ((z ∈ ran B → z ∈ dom A) ↔ (∃x xBz → ∃y zAy))
22	21	bial 695	. . 3 ⊢ (∀z(z ∈ ran B → z ∈ dom A) ↔ ∀z(∃x xBz → ∃y zAy))
23		visset 1350	. . . . . . 7 ⊢ x ∈ V
24	23	eldm2 2528	. . . . . 6 ⊢ (x ∈ dom (A ∘ B) ↔ ∃y⟨x, y⟩ ∈ (A ∘ B))
25		visset 1350	. . . . . . . 8 ⊢ y ∈ V
26	23, 25	opelco 2509	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (A ∘ B) ↔ ∃z(xBz ∧ zAy))
27	26	biex 733	. . . . . 6 ⊢ (∃y⟨x, y⟩ ∈ (A ∘ B) ↔ ∃y∃z(xBz ∧ zAy))
28		excom 728	. . . . . . 7 ⊢ (∃y∃z(xBz ∧ zAy) ↔ ∃z∃y(xBz ∧ zAy))
29		19.42v 966	. . . . . . . 8 ⊢ (∃y(xBz ∧ zAy) ↔ (xBz ∧ ∃y zAy))
30	29	biex 733	. . . . . . 7 ⊢ (∃z∃y(xBz ∧ zAy) ↔ ∃z(xBz ∧ ∃y zAy))
31	28, 30	bitr 151	. . . . . 6 ⊢ (∃y∃z(xBz ∧ zAy) ↔ ∃z(xBz ∧ ∃y zAy))
32	24, 27, 31	3bitr 155	. . . . 5 ⊢ (x ∈ dom (A ∘ B) ↔ ∃z(xBz ∧ ∃y zAy))
33	23	eldm 2527	. . . . 5 ⊢ (x ∈ dom B ↔ ∃z xBz)
34	32, 33	bibi12i 462	. . . 4 ⊢ ((x ∈ dom (A ∘ B) ↔ x ∈ dom B) ↔ (∃z(xBz ∧ ∃y zAy) ↔ ∃z xBz))
35	34	bial 695	. . 3 ⊢ (∀x(x ∈ dom (A ∘ B) ↔ x ∈ dom B) ↔ ∀x(∃z(xBz ∧ ∃y zAy) ↔ ∃z xBz))
36	17, 22, 35	3imtr4 192	. 2 ⊢ (∀z(z ∈ ran B → z ∈ dom A) → ∀x(x ∈ dom (A ∘ B) ↔ x ∈ dom B))
37		dfss2 1497	. 2 ⊢ (ran B ⊆ dom A ↔ ∀z(z ∈ ran B → z ∈ dom A))
38		dfcleq 1098	. 2 ⊢ (dom (A ∘ B) = dom B ↔ ∀x(x ∈ dom (A ∘ B) ↔ x ∈ dom B))
39	36, 37, 38	3imtr4 192	1 ⊢ (ran B ⊆ dom A → dom (A ∘ B) = dom B)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  dom cdm 2410  ran crn 2411   ∘ ccom 2414

This theorem is referenced by:  dmcoeq 2573  fco 2760

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429

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