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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 6719 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
2 | dfrel2 6092 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
4 | 3 | coeq2d 5771 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹)) |
5 | f1ocnv 6728 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
6 | f1ococnv2 6743 | . . 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 2780 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 I cid 5488 ◡ccnv 5588 ↾ cres 5591 ∘ ccom 5593 Rel wrel 5594 –1-1-onto→wf1o 6432 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 |
This theorem is referenced by: f1cocnv1 6746 f1ocnvfv1 7148 fcof1oinvd 7165 mapen 8928 mapfien 9167 hashfacen 14166 hashfacenOLD 14167 setcinv 17805 catcisolem 17825 symggrp 19008 f1omvdco2 19056 pf1mpf 21518 ufldom 23113 motgrp 26904 fmptco1f1o 30968 fcobij 31057 symgfcoeu 31351 pmtrcnel2 31359 cycpmconjslem1 31421 cycpmconjslem2 31422 reprpmtf1o 32606 subfacp1lem5 33146 ltrncoidN 38142 trlcoabs2N 38736 trlcoat 38737 trlcone 38742 cdlemg47 38750 tgrpgrplem 38763 tendoipl 38811 cdlemi2 38833 cdlemk2 38846 cdlemk4 38848 cdlemk8 38852 tendocnv 39035 dvhgrp 39121 cdlemn8 39218 dihopelvalcpre 39262 dssmap2d 41630 rngcinv 45539 rngcinvALTV 45551 ringcinv 45590 ringcinvALTV 45614 |
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