Theorem dmcosseq 5923

Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2158. (Revised by BTernaryTau, 23-Jun-2025.) Avoid ax-10 2142 and ax-12 2178. (Revised by TM, 31-Dec-2025.)

Assertion

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

Proof of Theorem dmcosseq

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

Step	Hyp	Ref	Expression
1		dmcoss 5920	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
2	1	a1i 11	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
3		ssel 3931	. . . . . . . 8 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴))
4		vex 3442	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	4	elrn 5840	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
6	4	eldm 5847	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
7	5, 6	imbi12i 350	. . . . . . . . 9 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8		breq1 5098	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑧𝐵𝑦))
9	8	19.8aw 2051	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
10	9	imim1i 63	. . . . . . . . . 10 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
11		pm3.2 469	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
12	11	eximdv 1917	. . . . . . . . . 10 ⊢ (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
13	10, 12	sylcom 30	. . . . . . . . 9 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
14	7, 13	sylbi 217	. . . . . . . 8 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
15	3, 14	syl 17	. . . . . . 7 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
16	15	eximdv 1917	. . . . . 6 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
17		breq2 5099	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑤))
18		breq1 5098	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑤𝐴𝑧))
19	17, 18	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧)))
20	19	excomimw 2044	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
21	16, 20	syl6 35	. . . . 5 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
22		vex 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
23		vex 3442	. . . . . . 7 ⊢ 𝑧 ∈ V
24	22, 23	opelco 5818	. . . . . 6 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
25	24	exbii 1848	. . . . 5 ⊢ (∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
26	21, 25	imbitrrdi 252	. . . 4 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵)))
27	22	eldm 5847	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
28	22	eldm2 5848	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
29	26, 27, 28	3imtr4g 296	. . 3 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵 → 𝑥 ∈ dom (𝐴 ∘ 𝐵)))
30	29	ssrdv 3943	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴 ∘ 𝐵))
31	2, 30	eqssd 3955	1 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3905  ⟨cop 4585   class class class wbr 5095  dom cdm 5623  ran crn 5624   ∘ ccom 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634

This theorem is referenced by:  dmcoeq  5926  cycpmconjv  33097  dmcoss3  38429  comptiunov2i  43679  dvsinax  45895  hoicvr  46530  fnresfnco  47026