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| Mirrors > Home > MPE Home > Th. List > fnssresb | Structured version Visualization version GIF version | ||
| Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.) |
| Ref | Expression |
|---|---|
| fnssresb | ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-fn 6564 | . 2 ⊢ ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵)) | |
| 2 | fnfun 6668 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 3 | 2 | funresd 6609 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun (𝐹 ↾ 𝐵)) |
| 4 | 3 | biantrurd 532 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵))) |
| 5 | ssdmres 6031 | . . . 4 ⊢ (𝐵 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐵) = 𝐵) | |
| 6 | fndm 6671 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 7 | 6 | sseq2d 4016 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
| 8 | 5, 7 | bitr3id 285 | . . 3 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
| 9 | 4, 8 | bitr3d 281 | . 2 ⊢ (𝐹 Fn 𝐴 → ((Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = 𝐵) ↔ 𝐵 ⊆ 𝐴)) |
| 10 | 1, 9 | bitrid 283 | 1 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
| 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-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-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-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: fnssres 6691 wrdred1hash 14599 rhmsscrnghm 20665 rngcrescrhm 20684 plyreres 26324 xrge0pluscn 33939 icoreresf 37353 fnbrafvb 47166 rngcrescrhmALTV 48196 |
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