Theorem fssrescdmd 7124

Description: Restriction of a function to a subclass of its domain as a function with domain and codomain. (Contributed by AV, 13-May-2025.)

Hypotheses

Ref	Expression
fssrescdmd.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fssrescdmd.c	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
fssrescdmd.d	⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)

Assertion

Ref	Expression
fssrescdmd	⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)

Proof of Theorem fssrescdmd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fssrescdmd.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 6707	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fssrescdmd.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
4	2, 3	fnssresd 6660	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
5		resima 6012	. . . 4 ⊢ ((𝐹 ↾ 𝐶) “ 𝐶) = (𝐹 “ 𝐶)
6		fssrescdmd.d	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)
7	5, 6	eqsstrid 3972	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷)
8	1	ffund 6711	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9	8	funresd 6580	. . . 4 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐶))
10	1	fdmd 6717	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	3, 10	sseqtrrd 3971	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐹)
12		ssdmres 6010	. . . . . . . 8 ⊢ (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶)
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
14		eqcom 2769	. . . . . . 7 ⊢ (dom (𝐹 ↾ 𝐶) = 𝐶 ↔ 𝐶 = dom (𝐹 ↾ 𝐶))
15	13, 14	bitrdi 290	. . . . . 6 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ 𝐶 = dom (𝐹 ↾ 𝐶)))
16	11, 15	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐶 = dom (𝐹 ↾ 𝐶))
17	16	eqimssd 3990	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ dom (𝐹 ↾ 𝐶))
18		funimass4 6946	. . . 4 ⊢ ((Fun (𝐹 ↾ 𝐶) ∧ 𝐶 ⊆ dom (𝐹 ↾ 𝐶)) → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
19	9, 17, 18	syl2anc 596	. . 3 ⊢ (𝜑 → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
20	7, 19	mpbid 235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷)
21		ffnfv 7116	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 ↔ ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
22	4, 20, 21	sylanbrc 595	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  dom cdm 5659   ↾ cres 5661   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is used by:  isubgruhgr  48792