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| Mirrors > Home > MPE Home > Th. List > f1ococnv1 | Structured version Visualization version GIF version | ||
| Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.) |
| Ref | Expression |
|---|---|
| f1ococnv1 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1orel 6851 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
| 2 | dfrel2 6209 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
| 3 | 1, 2 | sylib 218 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
| 4 | 3 | coeq2d 5873 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹)) |
| 5 | f1ocnv 6860 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
| 6 | f1ococnv2 6875 | . . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → (◡𝐹 ∘ ◡◡𝐹) = ( I ↾ 𝐴)) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = ( I ↾ 𝐴)) |
| 8 | 4, 7 | eqtr3d 2779 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 I cid 5577 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 –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: f1cocnv1 6878 f1ocnvfv1 7296 fcof1oinvd 7313 mapen 9181 mapfien 9448 hashfacen 14493 setcinv 18135 catcisolem 18155 symggrp 19418 f1omvdco2 19466 rngcinv 20637 ringcinv 20671 pf1mpf 22356 ufldom 23970 motgrp 28551 fmptco1f1o 32643 fcobij 32733 symgfcoeu 33102 pmtrcnel2 33110 cycpmconjslem1 33174 cycpmconjslem2 33175 reprpmtf1o 34641 subfacp1lem5 35189 ltrncoidN 40130 trlcoabs2N 40724 trlcoat 40725 trlcone 40730 cdlemg47 40738 tgrpgrplem 40751 tendoipl 40799 cdlemi2 40821 cdlemk2 40834 cdlemk4 40836 cdlemk8 40840 tendocnv 41023 dvhgrp 41109 cdlemn8 41206 dihopelvalcpre 41250 aks6d1c6lem5 42178 dssmap2d 44035 rngcinvALTV 48192 ringcinvALTV 48226 |
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