Theorem eqresfnbd 43031

Description: Property of being the restriction of a function. Note that this is closer to funssres 6580 than fnssres 6658. (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 6659	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) Fn 𝐴)
4		resss 6000	. . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
5	3, 4	jctir 529	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹))
6		fneq1 6626	. . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ↔ (𝐹 ↾ 𝐴) Fn 𝐴))
7		sseq1 3962	. . . 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 6636	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
11	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → Fun 𝐹)
12		funssres 6580	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅)
13	12	eqcomd 2769	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅))
14		fndm 6638	. . . . . . . 8 ⊢ (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴)
15	14	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → dom 𝑅 = 𝐴)
16	15	reseq2d 5978	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝐹 ↾ dom 𝑅) = (𝐹 ↾ 𝐴))
17	16	eqeq2d 2774	. . . . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ⊆ wss 3905  dom cdm 5661   ↾ cres 5663  Fun wfun 6530   Fn wfn 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-fun 6538  df-fn 6539

This theorem is used by: (None)