Theorem idresdmsscocnv1 6274

Description: A class postcomposed with its converse includes the identity relation restricted to its domain. (Contributed by BJ, 25-Oct-2026.)

Assertion

Ref	Expression
idresdmsscocnv1	⊢ ( I ↾ dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴)

Proof of Theorem idresdmsscocnv1

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

Step	Hyp	Ref	Expression
1		relres 6006	. 2 ⊢ Rel ( I ↾ dom 𝐴)
2		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
3	2	brresi 5989	. . . 4 ⊢ (𝑥( I ↾ dom 𝐴)𝑦 ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 I 𝑦))
4		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
5	4	eldm 5888	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑧 𝑥𝐴𝑧)
6		vex 3458	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
7	6, 4	brcnv 5866	. . . . . . . . 9 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧)
8	4, 6, 4	brcoi 5852	. . . . . . . . 9 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡𝐴𝑥) → 𝑥(◡𝐴 ∘ 𝐴)𝑥)
9	7, 8	sylan2br 607	. . . . . . . 8 ⊢ ((𝑥𝐴𝑧 ∧ 𝑥𝐴𝑧) → 𝑥(◡𝐴 ∘ 𝐴)𝑥)
10	9	anidms 577	. . . . . . 7 ⊢ (𝑥𝐴𝑧 → 𝑥(◡𝐴 ∘ 𝐴)𝑥)
11	10	exlimiv 1965	. . . . . 6 ⊢ (∃𝑧 𝑥𝐴𝑧 → 𝑥(◡𝐴 ∘ 𝐴)𝑥)
12	5, 11	sylbi 220	. . . . 5 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥(◡𝐴 ∘ 𝐴)𝑥)
13	2	ideq 5835	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
14		breq2 5110	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥(◡𝐴 ∘ 𝐴)𝑥 ↔ 𝑥(◡𝐴 ∘ 𝐴)𝑦))
15	13, 14	sylbi 220	. . . . . 6 ⊢ (𝑥 I 𝑦 → (𝑥(◡𝐴 ∘ 𝐴)𝑥 ↔ 𝑥(◡𝐴 ∘ 𝐴)𝑦))
16	15	biimpac 484	. . . . 5 ⊢ ((𝑥(◡𝐴 ∘ 𝐴)𝑥 ∧ 𝑥 I 𝑦) → 𝑥(◡𝐴 ∘ 𝐴)𝑦)
17	12, 16	sylan 592	. . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑥 I 𝑦) → 𝑥(◡𝐴 ∘ 𝐴)𝑦)
18	3, 17	sylbi 220	. . 3 ⊢ (𝑥( I ↾ dom 𝐴)𝑦 → 𝑥(◡𝐴 ∘ 𝐴)𝑦)
19		df-br 5107	. . 3 ⊢ (𝑥( I ↾ dom 𝐴)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴))
20		df-br 5107	. . 3 ⊢ (𝑥(◡𝐴 ∘ 𝐴)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∘ 𝐴))
21	18, 19, 20	3imtr3i 294	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ dom 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (◡𝐴 ∘ 𝐴))
22	1, 21	relssi 5768	1 ⊢ ( I ↾ dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2147   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106   I cid 5549  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657   ∘ ccom 5659

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 5252  ax-pr 5395

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-ral 3081  df-rex 3091  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 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667

This theorem is used by:  idresrnsscocnv2  6275