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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnresdmss | Structured version Visualization version GIF version |
Description: A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fnresdmss | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6480 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | fndm 6481 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | 2 | adantr 484 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 = 𝐴) |
4 | simpr 488 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
5 | 3, 4 | eqsstrd 3939 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → dom 𝐹 ⊆ 𝐵) |
6 | relssres 5892 | . 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹) | |
7 | 1, 5, 6 | syl2an2r 685 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐵) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ⊆ wss 3866 dom cdm 5551 ↾ cres 5553 Rel wrel 5556 Fn wfn 6375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-dm 5561 df-res 5563 df-fun 6382 df-fn 6383 |
This theorem is referenced by: (None) |
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