Theorem eqresfnbd 42633

Description: Property of being the restriction of a function. Note that this is closer to funssres 6546 than fnssres 6625. (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 6626	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) Fn 𝐴)
4		resss 5970	. . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
5	3, 4	jctir 520	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹))
6		fneq1 6593	. . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ↔ (𝐹 ↾ 𝐴) Fn 𝐴))
7		sseq1 3961	. . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 ⊆ 𝐹 ↔ (𝐹 ↾ 𝐴) ⊆ 𝐹))
8	6, 7	anbi12d 633	. . 3 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → ((𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹)))
9	5, 8	syl5ibrcom 247	. 2 ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))
10	1	fnfund 6603	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → Fun 𝐹)
12		funssres 6546	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅)
13	12	eqcomd 2743	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅))
14		fndm 6605	. . . . . . . 8 ⊢ (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → dom 𝑅 = 𝐴)
16	15	reseq2d 5948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝐹 ↾ dom 𝑅) = (𝐹 ↾ 𝐴))
17	16	eqeq2d 2748	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝑅 = (𝐹 ↾ dom 𝑅) ↔ 𝑅 = (𝐹 ↾ 𝐴)))
18	13, 17	imbitrid 244	. . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ 𝐴)))
19	11, 18	mpand 696	. . 3 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝑅 ⊆ 𝐹 → 𝑅 = (𝐹 ↾ 𝐴)))
20	19	expimpd 453	. 2 ⊢ (𝜑 → ((𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ 𝐴)))
21	9, 20	impbid 212	1 ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ⊆ wss 3903  dom cdm 5634   ↾ cres 5636  Fun wfun 6496   Fn wfn 6497

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-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

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-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-fun 6504  df-fn 6505

This theorem is referenced by: (None)