Theorem fssrescdmd 7071

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 6661	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fssrescdmd.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
4	2, 3	fnssresd 6614	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
5		resima 5972	. . . 4 ⊢ ((𝐹 ↾ 𝐶) “ 𝐶) = (𝐹 “ 𝐶)
6		fssrescdmd.d	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)
7	5, 6	eqsstrid 3961	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷)
8	1	ffund 6664	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9	8	funresd 6533	. . . 4 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐶))
10	1	fdmd 6670	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	3, 10	sseqtrrd 3960	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐹)
12		ssdmres 5970	. . . . . . . 8 ⊢ (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶)
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐶) = 𝐶))
14		eqcom 2744	. . . . . . 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 3979	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ dom (𝐹 ↾ 𝐶))
18		funimass4 6896	. . . 4 ⊢ ((Fun (𝐹 ↾ 𝐶) ∧ 𝐶 ⊆ dom (𝐹 ↾ 𝐶)) → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
19	9, 17, 18	syl2anc 585	. . 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 7063	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 ↔ ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
22	4, 20, 21	sylanbrc 584	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  dom cdm 5622   ↾ cres 5624   “ cima 5625  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498

This theorem is referenced by:  isubgruhgr  48341