Theorem cotrgOLD 6021

Description: Obsolete version of cotrg 6020 as of 19-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6022. (Revised by Richard Penner, 24-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cotrgOLD	⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧

Proof of Theorem cotrgOLD

Step	Hyp	Ref	Expression
1		relco 6019	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		ssrel 5694	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶)))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶))
4		vex 3437	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3437	. . . . . . . 8 ⊢ 𝑧 ∈ V
6	4, 5	opelco 5783	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧))
7		df-br 5076	. . . . . . . 8 ⊢ (𝑥𝐶𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐶)
8	7	bicomi 223	. . . . . . 7 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐶 ↔ 𝑥𝐶𝑧)
9	6, 8	imbi12i 351	. . . . . 6 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
10		19.23v 1946	. . . . . 6 ⊢ (∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ (∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
11	9, 10	bitr4i 277	. . . . 5 ⊢ ((⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
12	11	albii 1822	. . . 4 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
13		alcom 2157	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
14	12, 13	bitri 274	. . 3 ⊢ (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
15	14	albii 1822	. 2 ⊢ (∀𝑥∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) → ⟨𝑥, 𝑧⟩ ∈ 𝐶) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
16	3, 15	bitri 274	1 ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782   ∈ wcel 2107   ⊆ wss 3888  ⟨cop 4568   class class class wbr 5075   ∘ ccom 5594  Rel wrel 5595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5076  df-opab 5138  df-xp 5596  df-rel 5597  df-co 5599

This theorem is referenced by: (None)