Theorem dmcosseqOLDOLD 5926

Description: Obsolete version of dmcosseq 5924 as of 23-Jun-2025. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem dmcosseqOLDOLD

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

Step	Hyp	Ref	Expression
1		dmcoss 5921	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
2	1	a1i 11	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
3		ssel 3925	. . . . . . . 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		19.8a 2186	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
9	8	imim1i 63	. . . . . . . . . 10 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10		pm3.2 469	. . . . . . . . . . 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 217	. . . . . . . 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 2167	. . . . . 6 ⊢ (∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
17	15, 16	imbitrrdi 252	. . . . 5 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
18		vex 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
19		vex 3442	. . . . . . 7 ⊢ 𝑧 ∈ V
20	18, 19	opelco 5818	. . . . . 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	imbitrrdi 252	. . . 4 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵)))
23	18	eldm 5847	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
24	18	eldm2 5848	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
25	22, 23, 24	3imtr4g 296	. . 3 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵 → 𝑥 ∈ dom (𝐴 ∘ 𝐵)))
26	25	ssrdv 3937	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴 ∘ 𝐵))
27	2, 26	eqssd 3949	1 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ⊆ wss 3899  ⟨cop 4583   class class class wbr 5095  dom cdm 5621  ran crn 5622   ∘ ccom 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-12 2182  ax-ext 2705  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 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632

This theorem is referenced by: (None)