Theorem nvo 38221

Description: Two ways to express "sub-involutivity" of a class, by which is meant involutivity on the intersection of the domain and range (see nvo3 38223). (Contributed by BJ, 7-Nov-2026.)

Assertion

Ref	Expression
nvo	⊢ ((𝐴 ∘ 𝐴) ⊆ I ↔ (𝐴 ∘ 𝐴) = ( I ↾ (dom 𝐴 ∩ ran 𝐴)))

Proof of Theorem nvo

Step	Hyp	Ref	Expression
1		id 23	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ∘ 𝐴) ⊆ I )
2		cossxp 6296	. . . . . 6 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
3	2	a1i 11	. . . . 5 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴))
4	1, 3	ssind 4140	. . . 4 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ∘ 𝐴) ⊆ ( I ∩ (dom 𝐴 × ran 𝐴)))
5		idinxpres 6055	. . . 4 ⊢ ( I ∩ (dom 𝐴 × ran 𝐴)) = ( I ↾ (dom 𝐴 ∩ ran 𝐴))
6	4, 5	sseqtrdi 3953	. . 3 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ∘ 𝐴) ⊆ ( I ↾ (dom 𝐴 ∩ ran 𝐴)))
7		cossididres 38217	. . 3 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → ( I ↾ (dom 𝐴 ∩ ran 𝐴)) ⊆ (𝐴 ∘ 𝐴))
8	6, 7	eqssd 3930	. 2 ⊢ ((𝐴 ∘ 𝐴) ⊆ I → (𝐴 ∘ 𝐴) = ( I ↾ (dom 𝐴 ∩ ran 𝐴)))
9		id 23	. . 3 ⊢ ((𝐴 ∘ 𝐴) = ( I ↾ (dom 𝐴 ∩ ran 𝐴)) → (𝐴 ∘ 𝐴) = ( I ↾ (dom 𝐴 ∩ ran 𝐴)))
10		resss 6006	. . 3 ⊢ ( I ↾ (dom 𝐴 ∩ ran 𝐴)) ⊆ I
11	9, 10	eqsstrdi 3957	. 2 ⊢ ((𝐴 ∘ 𝐴) = ( I ↾ (dom 𝐴 ∩ ran 𝐴)) → (𝐴 ∘ 𝐴) ⊆ I )
12	8, 11	impbii 212	1 ⊢ ((𝐴 ∘ 𝐴) ⊆ I ↔ (𝐴 ∘ 𝐴) = ( I ↾ (dom 𝐴 ∩ ran 𝐴)))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ⊆ wss 3900   ∩ cin 4092   I cid 5550   × cxp 5654  dom cdm 5656  ran crn 5657   ↾ cres 5658   ∘ 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-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 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668

This theorem is used by: (None)