Theorem fnresdmss 44535

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 6650	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 6651	. . . 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 4016	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 ⊆ 𝐵)
6		relssres 6020	. 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)
7	1, 5, 6	syl2an2r 684	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ⊆ wss 3945  dom cdm 5672   ↾ cres 5674  Rel wrel 5677   Fn wfn 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-dm 5682  df-res 5684  df-fun 6544  df-fn 6545

This theorem is referenced by: (None)