Theorem dmcoss 5931

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 2147 and ax-12 2185. (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 1870	. . . . . 6 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
2		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	opelco 5827	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		breq2 5090	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧))
6	5	cbvexvw 2039	. . . . . 6 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
7	1, 4, 6	3imtr4i 292	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
8	7	eximi 1837	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦∃𝑦 𝑥𝐵𝑦)
9	5	exexw 2055	. . . 4 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦∃𝑦 𝑥𝐵𝑦)
10	8, 9	sylibr 234	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
11	2	eldm2 5857	. . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵))
12	2	eldm 5856	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
13	10, 11, 12	3imtr4i 292	. 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵)
14	13	ssriv 3926	1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵

Syntax hints:   ∧ wa 395  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3890  ⟨cop 4574   class class class wbr 5086  dom cdm 5631   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-co 5640  df-dm 5641

This theorem is referenced by:  rncoss  5933  dmcosseq  5934  dmcosseqOLD  5935  dmcosseqOLDOLD  5936  cossxp  6237  fvco4i  6942  cofunexg  7902  fin23lem30  10264  wunco  10656  relexpnndm  15003  mvdco  19420  f1omvdconj  19421  znleval  21534  ofco2  22416  tngtopn  24615  xppreima  32718  cycpmrn  33204  relexp0a  44143  dmtrclfvRP  44157  dmtposss  49345