Theorem dmcosseq

Description: Domain of a composition. (Contributed by NM, 28-May-1998) (Proof shortened by Andrew Salmon, 27-Aug-2011) Avoid ax-11 . (Revised by BTernaryTau, 23-Jun-2025)

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

Proof

Step	Hyp	Ref	Expression
1		dmcoss	⊢ dom ( 𝐴 ∘ 𝐵 ) ⊆ dom 𝐵
2	1	a1i	⊢ ( ran 𝐵 ⊆ dom 𝐴 → dom ( 𝐴 ∘ 𝐵 ) ⊆ dom 𝐵 )
3		ssel	⊢ ( ran 𝐵 ⊆ dom 𝐴 → ( 𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴 ) )
4		vex	⊢ 𝑦 ∈ V
5	4	elrn	⊢ ( 𝑦 ∈ ran 𝐵 ↔ ∃ 𝑥 𝑥 𝐵 𝑦 )
6	4	eldm	⊢ ( 𝑦 ∈ dom 𝐴 ↔ ∃ 𝑧 𝑦 𝐴 𝑧 )
7	5 6	imbi12i	⊢ ( ( 𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴 ) ↔ ( ∃ 𝑥 𝑥 𝐵 𝑦 → ∃ 𝑧 𝑦 𝐴 𝑧 ) )
8		19.8a	⊢ ( 𝑥 𝐵 𝑦 → ∃ 𝑥 𝑥 𝐵 𝑦 )
9	8	imim1i	⊢ ( ( ∃ 𝑥 𝑥 𝐵 𝑦 → ∃ 𝑧 𝑦 𝐴 𝑧 ) → ( 𝑥 𝐵 𝑦 → ∃ 𝑧 𝑦 𝐴 𝑧 ) )
10		pm3.2	⊢ ( 𝑥 𝐵 𝑦 → ( 𝑦 𝐴 𝑧 → ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
11	10	eximdv	⊢ ( 𝑥 𝐵 𝑦 → ( ∃ 𝑧 𝑦 𝐴 𝑧 → ∃ 𝑧 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
12	9 11	sylcom	⊢ ( ( ∃ 𝑥 𝑥 𝐵 𝑦 → ∃ 𝑧 𝑦 𝐴 𝑧 ) → ( 𝑥 𝐵 𝑦 → ∃ 𝑧 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
13	7 12	sylbi	⊢ ( ( 𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴 ) → ( 𝑥 𝐵 𝑦 → ∃ 𝑧 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
14	3 13	syl	⊢ ( ran 𝐵 ⊆ dom 𝐴 → ( 𝑥 𝐵 𝑦 → ∃ 𝑧 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
15	14	eximdv	⊢ ( ran 𝐵 ⊆ dom 𝐴 → ( ∃ 𝑦 𝑥 𝐵 𝑦 → ∃ 𝑦 ∃ 𝑧 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
16		breq2	⊢ ( 𝑦 = 𝑤 → ( 𝑥 𝐵 𝑦 ↔ 𝑥 𝐵 𝑤 ) )
17		breq1	⊢ ( 𝑦 = 𝑤 → ( 𝑦 𝐴 𝑧 ↔ 𝑤 𝐴 𝑧 ) )
18	16 17	anbi12d	⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ↔ ( 𝑥 𝐵 𝑤 ∧ 𝑤 𝐴 𝑧 ) ) )
19	18	excomimw	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → ∃ 𝑧 ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) )
20	15 19	syl6	⊢ ( ran 𝐵 ⊆ dom 𝐴 → ( ∃ 𝑦 𝑥 𝐵 𝑦 → ∃ 𝑧 ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) )
21		vex	⊢ 𝑥 ∈ V
22		vex	⊢ 𝑧 ∈ V
23	21 22	opelco	⊢ ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) ↔ ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) )
24	23	exbii	⊢ ( ∃ 𝑧 ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) ↔ ∃ 𝑧 ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) )
25	20 24	imbitrrdi	⊢ ( ran 𝐵 ⊆ dom 𝐴 → ( ∃ 𝑦 𝑥 𝐵 𝑦 → ∃ 𝑧 ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) ) )
26	21	eldm	⊢ ( 𝑥 ∈ dom 𝐵 ↔ ∃ 𝑦 𝑥 𝐵 𝑦 )
27	21	eldm2	⊢ ( 𝑥 ∈ dom ( 𝐴 ∘ 𝐵 ) ↔ ∃ 𝑧 ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) )
28	25 26 27	3imtr4g	⊢ ( ran 𝐵 ⊆ dom 𝐴 → ( 𝑥 ∈ dom 𝐵 → 𝑥 ∈ dom ( 𝐴 ∘ 𝐵 ) ) )
29	28	ssrdv	⊢ ( ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom ( 𝐴 ∘ 𝐵 ) )
30	2 29	eqssd	⊢ ( ran 𝐵 ⊆ dom 𝐴 → dom ( 𝐴 ∘ 𝐵 ) = dom 𝐵 )