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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 6610 than fnssres 6691. (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 6692 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) Fn 𝐴) |
| 4 | resss 6019 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
| 5 | 3, 4 | jctir 520 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ (𝐹 ↾ 𝐴) ⊆ 𝐹)) |
| 6 | fneq1 6659 | . . . 4 ⊢ (𝑅 = (𝐹 ↾ 𝐴) → (𝑅 Fn 𝐴 ↔ (𝐹 ↾ 𝐴) Fn 𝐴)) | |
| 7 | sseq1 4009 | . . . 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 6669 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) |
| 11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → Fun 𝐹) |
| 12 | funssres 6610 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → (𝐹 ↾ dom 𝑅) = 𝑅) | |
| 13 | 12 | eqcomd 2743 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑅 ⊆ 𝐹) → 𝑅 = (𝐹 ↾ dom 𝑅)) |
| 14 | fndm 6671 | . . . . . . . 8 ⊢ (𝑅 Fn 𝐴 → dom 𝑅 = 𝐴) | |
| 15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 Fn 𝐴) → dom 𝑅 = 𝐴) |
| 16 | 15 | reseq2d 5997 | . . . . . 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 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 1540 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: (None) |
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