Theorem dmcosseq 5809

Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Proof of Theorem dmcosseq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmcoss 5807	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
2	1	a1i 11	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
3		ssel 3908	. . . . . . . 8 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴))
4		vex 3444	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	4	elrn 5786	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
6	4	eldm 5733	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
7	5, 6	imbi12i 354	. . . . . . . . 9 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8		19.8a 2178	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
9	8	imim1i 63	. . . . . . . . . 10 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10		pm3.2 473	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
11	10	eximdv 1918	. . . . . . . . . 10 ⊢ (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
12	9, 11	sylcom 30	. . . . . . . . 9 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
13	7, 12	sylbi 220	. . . . . . . 8 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
14	3, 13	syl 17	. . . . . . 7 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
15	14	eximdv 1918	. . . . . 6 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
16		excom 2166	. . . . . 6 ⊢ (∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
17	15, 16	syl6ibr 255	. . . . 5 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
18		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
19		vex 3444	. . . . . . 7 ⊢ 𝑧 ∈ V
20	18, 19	opelco 5706	. . . . . 6 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
21	20	exbii 1849	. . . . 5 ⊢ (∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
22	17, 21	syl6ibr 255	. . . 4 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵)))
23	18	eldm 5733	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
24	18	eldm2 5734	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
25	22, 23, 24	3imtr4g 299	. . 3 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵 → 𝑥 ∈ dom (𝐴 ∘ 𝐵)))
26	25	ssrdv 3921	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴 ∘ 𝐵))
27	2, 26	eqssd 3932	1 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030  dom cdm 5519  ran crn 5520   ∘ ccom 5523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530

This theorem is referenced by:  dmcoeq  5810  fnco  6437  cycpmconjv  30834  dmcoss3  35853  comptiunov2i  40407  dvsinax  42555  hoicvr  43187  fnresfnco  43633