Theorem dmcoss 5956

Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2178 and ax-12 2215. (Revised by TM, 31-Dec-2025.)

Assertion

Ref	Expression
dmcoss	⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵

Proof of Theorem dmcoss

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

Step	Hyp	Ref	Expression
1		exsimpl 1891	. . . . . 6 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
2		vex 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	opelco 5848	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		breq2 5109	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧))
6	5	cbvexvw 2060	. . . . . 6 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
7	1, 4, 6	3imtr4i 295	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
8	7	eximi 1858	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦∃𝑦 𝑥𝐵𝑦)
9	5	exexw 2076	. . . 4 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦∃𝑦 𝑥𝐵𝑦)
10	8, 9	sylibr 237	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
11	2	eldm2 5882	. . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵))
12	2	eldm 5881	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
13	10, 11, 12	3imtr4i 295	. 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵)
14	13	ssriv 3943	1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵

Syntax hints:   ∧ wa 400  ∃wex 1802   ∈ wcel 2145   ⊆ wss 3907  ⟨cop 4591   class class class wbr 5105  dom cdm 5652   ∘ ccom 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-co 5661  df-dm 5662

This theorem is referenced by:  rncoss  5958  dmcosseq  5959  dmcosseqOLD  5960  cossxp  6263  fvco4i  6973  cofunexg  7934  fin23lem30  10314  wunco  10706  relexpnndm  15068  mvdco  19506  f1omvdconj  19507  znleval  21664  ofco2  22569  tngtopn  24768  xppreima  32902  cycpmrn  33376  relexp0a  44304  dmtrclfvRP  44318  dmtposss  49505