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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqresfnbd | Structured version Visualization version GIF version |
Description: Property of being the restriction of a function. Note that this is closer to funssres 6612 than fnssres 6692. (Contributed by SN, 11-Mar-2025.) |
Ref | Expression |
---|---|
eqresfnbd.g | ⊢ (𝜑 → 𝐹 Fn 𝐵) |
eqresfnbd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
eqresfnbd | ⊢ (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqresfnbd.g | . . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐵) | |
2 | eqresfnbd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | 1, 2 | fnssresd 6693 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) Fn 𝐴) |
4 | resss 6022 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
5 | 3, 4 | jctir 520 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹)) |
6 | fneq1 6660 | . . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ↔ (𝐹 ↾ 𝐴) Fn 𝐴)) | |
7 | sseq1 4021 | . . . 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 6670 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → Fun 𝐹) |
12 | funssres 6612 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅) | |
13 | 12 | eqcomd 2741 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅)) |
14 | fndm 6672 | . . . . . . . 8 ⊢ (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴) | |
15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → dom 𝑅 = 𝐴) |
16 | 15 | reseq2d 6000 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → (𝐹 ↾ dom 𝑅) = (𝐹 ↾ 𝐴)) |
17 | 16 | eqeq2d 2746 | . . . . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3963 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 Fn wfn 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-fun 6565 df-fn 6566 |
This theorem is referenced by: (None) |
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