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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 6569 | . . 3 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | funcnvres 6580 | . . 3 ⊢ (Fun ◡◡𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (◡◡𝐹 ↾ (◡𝐹 “ 𝐴))) |
4 | funrel 6519 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | dfrel2 6142 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹) |
7 | 6 | reseq1d 5937 | . 2 ⊢ (Fun 𝐹 → (◡◡𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
8 | 3, 7 | eqtrd 2773 | 1 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ◡ccnv 5633 ↾ cres 5636 “ cima 5637 Rel wrel 5639 Fun wfun 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 |
This theorem is referenced by: funimacnv 6583 foimacnv 6802 unbenlem 16785 ofco2 21816 dvlog 26022 fresf1o 31591 fressupp 31649 |
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