Theorem fssrescdmd 7065

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 6657	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fssrescdmd.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
4	2, 3	fnssresd 6610	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
5		resima 5968	. . . 4 ⊢ ((𝐹 ↾ 𝐶) “ 𝐶) = (𝐹 “ 𝐶)
6		fssrescdmd.d	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)
7	5, 6	eqsstrid 3969	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷)
8	1	ffund 6660	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9	8	funresd 6529	. . . 4 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐶))
10	1	fdmd 6666	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	3, 10	sseqtrrd 3968	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐹)
12		ssdmres 5966	. . . . . . . 8 ⊢ (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶)
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
14		eqcom 2740	. . . . . . 7 ⊢ (dom (𝐹 ↾ 𝐶) = 𝐶 ↔ 𝐶 = dom (𝐹 ↾ 𝐶))
15	13, 14	bitrdi 287	. . . . . 6 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ 𝐶 = dom (𝐹 ↾ 𝐶)))
16	11, 15	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝐶 = dom (𝐹 ↾ 𝐶))
17	16	eqimssd 3987	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ dom (𝐹 ↾ 𝐶))
18		funimass4 6892	. . . 4 ⊢ ((Fun (𝐹 ↾ 𝐶) ∧ 𝐶 ⊆ dom (𝐹 ↾ 𝐶)) → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
19	9, 17, 18	syl2anc 584	. . 3 ⊢ (𝜑 → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
20	7, 19	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷)
21		ffnfv 7058	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 ↔ ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
22	4, 20, 21	sylanbrc 583	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  dom cdm 5619   ↾ cres 5621   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494

This theorem is referenced by:  isubgruhgr  47992