Theorem fnresdmss 45146

Description: A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
fnresdmss	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)

Proof of Theorem fnresdmss

Step	Hyp	Ref	Expression
1		fnrel 6584	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 6585	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 = 𝐴)
4		simpr 484	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
5	3, 4	eqsstrd 3970	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 ⊆ 𝐵)
6		relssres 5973	. 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)
7	1, 5, 6	syl2an2r 685	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ⊆ wss 3903  dom cdm 5619   ↾ cres 5621  Rel wrel 5624   Fn wfn 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-dm 5629  df-res 5631  df-fun 6484  df-fn 6485

This theorem is referenced by: (None)