![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funcnvres | Structured version Visualization version GIF version |
Description: The converse of a restricted function. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
funcnvres | ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5279 | . . . 4 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
2 | df-rn 5277 | . . . 4 ⊢ ran (𝐹 ↾ 𝐴) = dom ◡(𝐹 ↾ 𝐴) | |
3 | 1, 2 | eqtri 2782 | . . 3 ⊢ (𝐹 “ 𝐴) = dom ◡(𝐹 ↾ 𝐴) |
4 | 3 | reseq2i 5548 | . 2 ⊢ (◡𝐹 ↾ (𝐹 “ 𝐴)) = (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) |
5 | resss 5580 | . . . 4 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹 | |
6 | cnvss 5450 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 |
8 | funssres 6091 | . . 3 ⊢ ((Fun ◡𝐹 ∧ ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹) → (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) = ◡(𝐹 ↾ 𝐴)) | |
9 | 7, 8 | mpan2 709 | . 2 ⊢ (Fun ◡𝐹 → (◡𝐹 ↾ dom ◡(𝐹 ↾ 𝐴)) = ◡(𝐹 ↾ 𝐴)) |
10 | 4, 9 | syl5req 2807 | 1 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ⊆ wss 3715 ◡ccnv 5265 dom cdm 5266 ran crn 5267 ↾ cres 5268 “ cima 5269 Fun wfun 6043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-fun 6051 |
This theorem is referenced by: cnvresid 6129 funcnvres2 6130 f1orescnv 6314 f1imacnv 6315 sbthlem4 8240 fpwwe2lem6 9669 fpwwe2lem9 9672 hmeores 21796 dvcnvrelem2 24000 dfrelog 24532 efopnlem2 24623 diophrw 37842 |
Copyright terms: Public domain | W3C validator |