Theorem dmcoss 5949

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 2174 and ax-12 2211. (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 1887	. . . . . 6 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
2		vex 3457	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3457	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	opelco 5841	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		breq2 5103	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧))
6	5	cbvexvw 2056	. . . . . 6 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
7	1, 4, 6	3imtr4i 294	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
8	7	eximi 1854	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦∃𝑦 𝑥𝐵𝑦)
9	5	exexw 2072	. . . 4 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑦∃𝑦 𝑥𝐵𝑦)
10	8, 9	sylibr 236	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
11	2	eldm2 5875	. . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵))
12	2	eldm 5874	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
13	10, 11, 12	3imtr4i 294	. 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵)
14	13	ssriv 3940	1 ⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵

Syntax hints:   ∧ wa 399  ∃wex 1798   ∈ wcel 2141   ⊆ wss 3904  ⟨cop 4587   class class class wbr 5099  dom cdm 5645   ∘ ccom 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-co 5654  df-dm 5655

This theorem is referenced by:  rncoss  5951  dmcosseq  5952  dmcosseqOLD  5953  dmcosseqOLDOLD  5954  cossxp  6253  fvco4i  6963  cofunexg  7924  fin23lem30  10294  wunco  10686  relexpnndm  15049  mvdco  19466  f1omvdconj  19467  znleval  21584  ofco2  22489  tngtopn  24688  xppreima  32795  cycpmrn  33282  relexp0a  44245  dmtrclfvRP  44259  dmtposss  49450