Theorem eqresfnbd 42213

Description: Property of being the restriction of a function. Note that this is closer to funssres 6577 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 5986	. . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
5	3, 4	jctir 520	. . 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 3982	. . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 ⊆ 𝐹 ↔ (𝐹 ↾ 𝐴) ⊆ 𝐹))
8	6, 7	anbi12d 632	. . 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 6636	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → Fun 𝐹)
12		funssres 6577	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅)
13	12	eqcomd 2740	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅))
14		fndm 6638	. . . . . . . 8 ⊢ (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → dom 𝑅 = 𝐴)
16	15	reseq2d 5964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝐹 ↾ dom 𝑅) = (𝐹 ↾ 𝐴))
17	16	eqeq2d 2745	. . . . 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 695	. . 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 1539   ⊆ wss 3924  dom cdm 5652   ↾ cres 5654  Fun wfun 6522   Fn wfn 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5118  df-opab 5180  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-res 5664  df-fun 6530  df-fn 6531

This theorem is referenced by: (None)