Theorem dmcosseqOLD 5928

Description: Obsolete version of dmcosseq 5927 as of 31-Dec-2025. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2170. (Revised by BTernaryTau, 23-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem dmcosseqOLD

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

Step	Hyp	Ref	Expression
1		dmcoss 5924	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
2	1	a1i 11	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
3		ssel 3911	. . . . . . . 8 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴))
4		vex 3437	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	4	elrn 5842	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
6	4	eldm 5849	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
7	5, 6	imbi12i 352	. . . . . . . . 9 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8		19.8a 2195	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
9	8	imim1i 63	. . . . . . . . . 10 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10		pm3.2 471	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
11	10	eximdv 1925	. . . . . . . . . 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 219	. . . . . . . 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 1925	. . . . . 6 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
16		breq2 5079	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑤))
17		breq1 5078	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦𝐴𝑧 ↔ 𝑤𝐴𝑧))
18	16, 17	anbi12d 639	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑧)))
19	18	excomimw 2052	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
20	15, 19	syl6 35	. . . . 5 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
21		vex 3437	. . . . . . 7 ⊢ 𝑥 ∈ V
22		vex 3437	. . . . . . 7 ⊢ 𝑧 ∈ V
23	21, 22	opelco 5816	. . . . . 6 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
24	23	exbii 1856	. . . . 5 ⊢ (∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
25	20, 24	imbitrrdi 254	. . . 4 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵)))
26	21	eldm 5849	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
27	21	eldm2 5850	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
28	25, 26, 27	3imtr4g 298	. . 3 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵 → 𝑥 ∈ dom (𝐴 ∘ 𝐵)))
29	28	ssrdv 3923	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴 ∘ 𝐵))
30	2, 29	eqssd 3934	1 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ⊆ wss 3885  ⟨cop 4564   class class class wbr 5075  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 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632

This theorem is referenced by: (None)