Theorem eqresfnbd 42927

Description: Property of being the restriction of a function. Note that this is closer to funssres 6581 than fnssres 6659. (Contributed by SN, 11-Mar-2025.)

Hypotheses

Ref	Expression
eqresfnbd.g	⊢ (𝜑 → 𝐹 Fn 𝐵)
eqresfnbd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
eqresfnbd	⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))

Proof of Theorem eqresfnbd

Step	Hyp	Ref	Expression
1		eqresfnbd.g	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐵)
2		eqresfnbd.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3	1, 2	fnssresd 6660	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) Fn 𝐴)
4		resss 6001	. . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
5	3, 4	jctir 529	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹))
6		fneq1 6627	. . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ↔ (𝐹 ↾ 𝐴) Fn 𝐴))
7		sseq1 3970	. . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 ⊆ 𝐹 ↔ (𝐹 ↾ 𝐴) ⊆ 𝐹))
8	6, 7	anbi12d 643	. . 3 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → ((𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹)))
9	5, 8	syl5ibrcom 250	. 2 ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))
10	1	fnfund 6637	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
11	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → Fun 𝐹)
12		funssres 6581	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅)
13	12	eqcomd 2775	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅))
14		fndm 6639	. . . . . . . 8 ⊢ (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴)
15	14	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → dom 𝑅 = 𝐴)
16	15	reseq2d 5979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝐹 ↾ dom 𝑅) = (𝐹 ↾ 𝐴))
17	16	eqeq2d 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝑅 = (𝐹 ↾ dom 𝑅) ↔ 𝑅 = (𝐹 ↾ 𝐴)))
18	13, 17	imbitrid 247	. . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ 𝐴)))
19	11, 18	mpand 707	. . 3 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝑅 ⊆ 𝐹 → 𝑅 = (𝐹 ↾ 𝐴)))
20	19	expimpd 458	. 2 ⊢ (𝜑 → ((𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ 𝐴)))
21	9, 20	impbid 215	1 ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ⊆ wss 3913  dom cdm 5662   ↾ cres 5664  Fun wfun 6531   Fn wfn 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-fun 6539  df-fn 6540

This theorem is referenced by: (None)