Theorem cocnv2ss 38206

Description: Characterizing when the composition of a class with a converse is included in another. (Contributed by BJ, 22-Oct-2026.)

Assertion

Ref	Expression
cocnv2ss	⊢ ((𝐵 ∘ ◡𝐴) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑦𝐴𝑥 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧))

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

Proof of Theorem cocnv2ss

Step	Hyp	Ref	Expression
1		cotrg 6121	. 2 ⊢ ((𝐵 ∘ ◡𝐴) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥◡𝐴𝑦 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧))
2		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	brcnv 5870	. . . . 5 ⊢ (𝑥◡𝐴𝑦 ↔ 𝑦𝐴𝑥)
5	4	anbi1i 636	. . . 4 ⊢ ((𝑥◡𝐴𝑦 ∧ 𝑦𝐵𝑧) ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑧))
6	5	imbi1i 352	. . 3 ⊢ (((𝑥◡𝐴𝑦 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧) ↔ ((𝑦𝐴𝑥 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧))
7	6	3albii 1854	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥◡𝐴𝑦 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦𝐴𝑥 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧))
8	1, 7	bitri 278	1 ⊢ ((𝐵 ∘ ◡𝐴) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑦𝐴𝑥 ∧ 𝑦𝐵𝑧) → 𝑥𝐶𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ⊆ wss 3900   class class class wbr 5106  ^◡ccnv 5655   ∘ ccom 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665

This theorem is used by:  cocnv2ssid  38207