Theorem fssrescdmd 7075

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 6663	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fssrescdmd.c	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
4	2, 3	fnssresd 6616	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
5		resima 5974	. . . 4 ⊢ ((𝐹 ↾ 𝐶) “ 𝐶) = (𝐹 “ 𝐶)
6		fssrescdmd.d	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)
7	5, 6	eqsstrid 3960	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷)
8	1	ffund 6666	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
9	8	funresd 6535	. . . 4 ⊢ (𝜑 → Fun (𝐹 ↾ 𝐶))
10	1	fdmd 6672	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	3, 10	sseqtrrd 3959	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ dom 𝐹)
12		ssdmres 5972	. . . . . . . 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 288	. . . . . 6 ⊢ (𝜑 → (𝐶 ⊆ dom 𝐹 ↔ 𝐶 = dom (𝐹 ↾ 𝐶)))
16	11, 15	mpbid 233	. . . . 5 ⊢ (𝜑 → 𝐶 = dom (𝐹 ↾ 𝐶))
17	16	eqimssd 3978	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ dom (𝐹 ↾ 𝐶))
18		funimass4 6898	. . . 4 ⊢ ((Fun (𝐹 ↾ 𝐶) ∧ 𝐶 ⊆ dom (𝐹 ↾ 𝐶)) → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
19	9, 17, 18	syl2anc 590	. . 3 ⊢ (𝜑 → (((𝐹 ↾ 𝐶) “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
20	7, 19	mpbid 233	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷)
21		ffnfv 7067	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐷 ↔ ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ∀𝑥 ∈ 𝐶 ((𝐹 ↾ 𝐶)‘𝑥) ∈ 𝐷))
22	4, 20, 21	sylanbrc 589	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ⊆ wss 3890  dom cdm 5625   ↾ cres 5627   “ cima 5628  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  isubgruhgr  48360