Theorem dmcosseq 4603

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 4601	. . 3 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵
2	1	a1i 9	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵)
3		ssel 2939	. . . . . . . 8 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴))
4		vex 2560	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	4	elrn 4577	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
6	4	eldm 4532	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
7	5, 6	imbi12i 228	. . . . . . . . 9 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8		19.8a 1482	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
9	8	imim1i 54	. . . . . . . . . 10 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10		pm3.2 126	. . . . . . . . . . 11 ⊢ (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
11	10	eximdv 1760	. . . . . . . . . 10 ⊢ (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
12	9, 11	sylcom 25	. . . . . . . . 9 ⊢ ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
13	7, 12	sylbi 114	. . . . . . . 8 ⊢ ((𝑦 ∈ ran 𝐵 → 𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
14	3, 13	syl 14	. . . . . . 7 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
15	14	eximdv 1760	. . . . . 6 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
16		excom 1554	. . . . . 6 ⊢ (∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) ↔ ∃𝑦∃𝑧(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
17	15, 16	syl6ibr 151	. . . . 5 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧)))
18		vex 2560	. . . . . . 7 ⊢ 𝑥 ∈ V
19		vex 2560	. . . . . . 7 ⊢ 𝑧 ∈ V
20	18, 19	opelco 4507	. . . . . 6 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
21	20	exbii 1496	. . . . 5 ⊢ (∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
22	17, 21	syl6ibr 151	. . . 4 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵)))
23	18	eldm 4532	. . . 4 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
24	18	eldm2 4533	. . . 4 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑧⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
25	22, 23, 24	3imtr4g 194	. . 3 ⊢ (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵 → 𝑥 ∈ dom (𝐴 ∘ 𝐵)))
26	25	ssrdv 2951	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴 ∘ 𝐵))
27	2, 26	eqssd 2962	1 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  dom cdm 4345  ran crn 4346   ∘ ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356

This theorem is referenced by:  dmcoeq  4604  fnco  5007