Theorem fnresdmss 45629

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 6591	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 6592	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	adantr 482	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 = 𝐴)
4		simpr 486	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
5	3, 4	eqsstrd 3951	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 ⊆ 𝐵)
6		relssres 5981	. 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)
7	1, 5, 6	syl2an2r 692	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ⊆ wss 3885  dom cdm 5621   ↾ cres 5623  Rel wrel 5626   Fn wfn 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-dm 5631  df-res 5633  df-fun 6491  df-fn 6492

This theorem is referenced by: (None)