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| Mirrors > Home > MPE Home > Th. List > funcnvres2 | Structured version Visualization version 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 6633 | . . 3 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
| 2 | funcnvres 6644 | . . 3 ⊢ (Fun ◡◡𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) |
| 4 | funrel 6583 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 5 | dfrel2 6209 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
| 6 | 4, 5 | sylib 218 | . . 3 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
| 7 | 6 | reseq1d 5996 | . 2 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
| 8 | 3, 7 | eqtrd 2777 | 1 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 Rel wrel 5690 Fun wfun 6555 |
| 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-12 2177 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-mo 2540 df-eu 2569 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-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: funimacnv 6647 foimacnv 6865 unbenlem 16946 ofco2 22457 dvlog 26693 fresf1o 32641 fressupp 32697 |
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