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Mirrors > Home > ILE Home > Th. List > funcnvres2 | GIF version |
Description: The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.) |
Ref | Expression |
---|---|
funcnvres2 | ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnvcnv 4958 | . . 3 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | funcnvres 4972 | . . 3 ⊢ (Fun ◡◡𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) |
4 | funrel 4919 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | dfrel2 4771 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
6 | 4, 5 | sylib 127 | . . 3 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
7 | 6 | reseq1d 4611 | . 2 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
8 | 3, 7 | eqtrd 2072 | 1 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ◡ccnv 4344 ↾ cres 4347 “ cima 4348 Rel wrel 4350 Fun wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 |
This theorem is referenced by: funimacnv 4975 foimacnv 5144 |
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