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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 6837 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
2 | dfrel2 6189 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
4 | 3 | coeq2d 5863 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹)) |
5 | f1ocnv 6846 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
6 | f1ococnv2 6861 | . . 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 2775 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 I cid 5574 ◡ccnv 5676 ↾ cres 5679 ∘ ccom 5681 Rel wrel 5682 –1-1-onto→wf1o 6543 |
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-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 |
This theorem is referenced by: f1cocnv1 6864 f1ocnvfv1 7274 fcof1oinvd 7291 mapen 9141 mapfien 9403 hashfacen 14413 hashfacenOLD 14414 setcinv 18040 catcisolem 18060 symggrp 19268 f1omvdco2 19316 pf1mpf 21871 ufldom 23466 motgrp 27794 fmptco1f1o 31857 fcobij 31947 symgfcoeu 32243 pmtrcnel2 32251 cycpmconjslem1 32313 cycpmconjslem2 32314 reprpmtf1o 33638 subfacp1lem5 34175 ltrncoidN 38999 trlcoabs2N 39593 trlcoat 39594 trlcone 39599 cdlemg47 39607 tgrpgrplem 39620 tendoipl 39668 cdlemi2 39690 cdlemk2 39703 cdlemk4 39705 cdlemk8 39709 tendocnv 39892 dvhgrp 39978 cdlemn8 40075 dihopelvalcpre 40119 dssmap2d 42773 rngcinv 46879 rngcinvALTV 46891 ringcinv 46930 ringcinvALTV 46954 |
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