Theorem dmcossOLD 5950

Description: Obsolete version of dmcosseq 5952 as of 31-Dec-2025. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem dmcossOLD

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

Step	Hyp	Ref	Expression
1		nfe1 2183	. . . 4 ⊢ Ⅎ𝑦∃𝑦 𝑥𝐵𝑦
2		exsimpl 1887	. . . . 5 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
3		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	opelco 5841	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6		breq2 5103	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝑧))
7	6	cbvexvw 2056	. . . . 5 ⊢ (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
8	2, 5, 7	3imtr4i 294	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
9	1, 8	exlimi 2251	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵) → ∃𝑦 𝑥𝐵𝑦)
10	3	eldm2 5875	. . 3 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ 𝐵))
11	3	eldm 5874	. . 3 ⊢ (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
12	9, 10, 11	3imtr4i 294	. 2 ⊢ (𝑥 ∈ dom (𝐴 ∘ 𝐵) → 𝑥 ∈ dom 𝐵)
13	12	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-10 2174  ax-12 2211  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-nf 1803  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: (None)