Theorem fnresdmss 45162

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 6620	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 6621	. . . 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 3981	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 ⊆ 𝐵)
6		relssres 5993	. 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 3914  dom cdm 5638   ↾ cres 5640  Rel wrel 5643   Fn wfn 6506

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-res 5650  df-fun 6513  df-fn 6514

This theorem is referenced by: (None)