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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 5241 | . . 3 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | funcnvres 5255 | . . 3 ⊢ (Fun ◡◡𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) |
4 | funrel 5199 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | dfrel2 5048 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
6 | 4, 5 | sylib 121 | . . 3 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
7 | 6 | reseq1d 4877 | . 2 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
8 | 3, 7 | eqtrd 2197 | 1 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ◡ccnv 4597 ↾ cres 4600 “ cima 4601 Rel wrel 4603 Fun wfun 5176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 |
This theorem is referenced by: funimacnv 5258 foimacnv 5444 |
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