Theorem fssrescdmd 7160

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 6748	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fssrescdmd.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
4	2, 3	fnssresd 6704	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
5		resima 6044	. . . 4 ⊢ ((𝐹 ↾ 𝐶) “ 𝐶) = (𝐹 “ 𝐶)
6		fssrescdmd.d	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)
7	5, 6	eqsstrid 4057	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷)
8	1	ffund 6751	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9	8	funresd 6621	. . . 4 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐶))
10	1	fdmd 6757	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	3, 10	sseqtrrd 4050	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐹)
12		ssdmres 6042	. . . . . . . 8 ⊢ (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶)
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
14		eqcom 2747	. . . . . . 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 4065	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ dom (𝐹 ↾ 𝐶))
18		funimass4 6986	. . . 4 ⊢ ((Fun (𝐹 ↾ 𝐶) ∧ 𝐶 ⊆ dom (𝐹 ↾ 𝐶)) → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
19	9, 17, 18	syl2anc 583	. . 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 7153	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 ↔ ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
22	4, 20, 21	sylanbrc 582	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  dom cdm 5700   ↾ cres 5702   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  isubgruhgr  47738