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| Mirrors > Home > MPE Home > Th. List > f1ococnv2 | Structured version Visualization version GIF version | ||
| Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.) |
| Ref | Expression |
|---|---|
| f1ococnv2 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1ofo 6855 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | |
| 2 | fococnv2 6874 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 I cid 5577 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 –onto→wfo 6559 –1-1-onto→wf1o 6560 |
| 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-10 2141 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-nf 1784 df-sb 2065 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-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: f1ococnv1 6877 f1ocnvfv2 7297 mapen 9181 hashfacen 14493 setcinv 18135 catcisolem 18155 symginv 19420 f1omvdco2 19466 gsumval3 19925 gsumzf1o 19930 rngcinv 20637 ringcinv 20671 psrass1lem 21952 evl1var 22340 pf1ind 22359 fcobij 32733 symgfcoeu 33102 cycpmconjvlem 33161 cycpmconjs 33176 cyc3conja 33177 erdsze2lem2 35209 ltrncoidN 40130 cdlemg46 40737 cdlemk45 40949 cdlemk55a 40961 tendocnv 41023 eldioph2 42773 rngcinvALTV 48192 ringcinvALTV 48226 |
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