Theorem comodular 38200

Description: A strong modular law for the composition of relations. The modular law is an axiom defining allegories. (Contributed by BJ, 7-Nov-2026.)

Assertion

Ref	Expression
comodular	⊢ ((𝐴 ∘ 𝐵) ∩ 𝐶) ⊆ ((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))

Proof of Theorem comodular

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

Step	Hyp	Ref	Expression
1		relco 6120	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relin1 5797	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → Rel ((𝐴 ∘ 𝐵) ∩ 𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ Rel ((𝐴 ∘ 𝐵) ∩ 𝐶)
4		brin 5161	. . . 4 ⊢ (𝑥((𝐴 ∘ 𝐵) ∩ 𝐶)𝑦 ↔ (𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥𝐶𝑦))
5		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
6		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
7	5, 6	brco 5858	. . . . . 6 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
8		simprl 783	. . . . . . . . . 10 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑥𝐵𝑧)
9		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑥𝐶𝑦)
10		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑧𝐴𝑦)
11		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
12	6, 11	brcnv 5870	. . . . . . . . . . . 12 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
13	10, 12	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑦◡𝐴𝑧)
14	5, 6, 11	brcoi 5856	. . . . . . . . . . 11 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦◡𝐴𝑧) → 𝑥(◡𝐴 ∘ 𝐶)𝑧)
15	9, 13, 14	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑥(◡𝐴 ∘ 𝐶)𝑧)
16		brin 5161	. . . . . . . . . 10 ⊢ (𝑥(𝐵 ∩ (◡𝐴 ∘ 𝐶))𝑧 ↔ (𝑥𝐵𝑧 ∧ 𝑥(◡𝐴 ∘ 𝐶)𝑧))
17	8, 15, 16	sylanbrc 595	. . . . . . . . 9 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑥(𝐵 ∩ (◡𝐴 ∘ 𝐶))𝑧)
18	11, 5	brcnv 5870	. . . . . . . . . . . 12 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧)
19	8, 18	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑧◡𝐵𝑥)
20	11, 5, 6	brcoi 5856	. . . . . . . . . . 11 ⊢ ((𝑧◡𝐵𝑥 ∧ 𝑥𝐶𝑦) → 𝑧(𝐶 ∘ ◡𝐵)𝑦)
21	19, 9, 20	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑧(𝐶 ∘ ◡𝐵)𝑦)
22		brin 5161	. . . . . . . . . 10 ⊢ (𝑧(𝐴 ∩ (𝐶 ∘ ◡𝐵))𝑦 ↔ (𝑧𝐴𝑦 ∧ 𝑧(𝐶 ∘ ◡𝐵)𝑦))
23	10, 21, 22	sylanbrc 595	. . . . . . . . 9 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑧(𝐴 ∩ (𝐶 ∘ ◡𝐵))𝑦)
24	5, 11, 6	brcoi 5856	. . . . . . . . 9 ⊢ ((𝑥(𝐵 ∩ (◡𝐴 ∘ 𝐶))𝑧 ∧ 𝑧(𝐴 ∩ (𝐶 ∘ ◡𝐵))𝑦) → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦)
25	17, 23, 24	syl2anc 596	. . . . . . . 8 ⊢ ((𝑥𝐶𝑦 ∧ (𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)) → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦)
26	25	ex 418	. . . . . . 7 ⊢ (𝑥𝐶𝑦 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦))
27	26	exlimdv 1968	. . . . . 6 ⊢ (𝑥𝐶𝑦 → (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦))
28	7, 27	biimtrid 245	. . . . 5 ⊢ (𝑥𝐶𝑦 → (𝑥(𝐴 ∘ 𝐵)𝑦 → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦))
29	28	impcom 413	. . . 4 ⊢ ((𝑥(𝐴 ∘ 𝐵)𝑦 ∧ 𝑥𝐶𝑦) → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦)
30	4, 29	sylbi 220	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ∩ 𝐶)𝑦 → 𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦)
31		df-br 5107	. . 3 ⊢ (𝑥((𝐴 ∘ 𝐵) ∩ 𝐶)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∩ 𝐶))
32		df-br 5107	. . 3 ⊢ (𝑥((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶))))
33	30, 31, 32	3imtr3i 294	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∘ 𝐵) ∩ 𝐶) → ⟨𝑥, 𝑦⟩ ∈ ((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶))))
34	3, 33	relssi 5770	1 ⊢ ((𝐴 ∘ 𝐵) ∩ 𝐶) ⊆ ((𝐴 ∩ (𝐶 ∘ ◡𝐵)) ∘ (𝐵 ∩ (◡𝐴 ∘ 𝐶)))

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  ⟨cop 4838   class class class wbr 5106  ^◡ccnv 5655   ∘ ccom 5660  Rel wrel 5661

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:  coineq0  38201  coindir  38210  coindi  38212