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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 6621 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
2 | dfrel2 6021 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | 1, 2 | sylib 221 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
4 | 3 | coeq2d 5705 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹)) |
5 | f1ocnv 6630 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
6 | f1ococnv2 6644 | . . 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 5428 ◡ccnv 5524 ↾ cres 5527 ∘ ccom 5529 Rel wrel 5530 –1-1-onto→wf1o 6338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 |
This theorem is referenced by: f1cocnv1 6647 f1ocnvfv1 7044 fcof1oinvd 7060 mapen 8731 mapfien 8945 hashfacen 13904 hashfacenOLD 13905 setcinv 17462 catcisolem 17482 symggrp 18646 f1omvdco2 18694 pf1mpf 21122 ufldom 22713 motgrp 26489 fmptco1f1o 30542 fcobij 30632 symgfcoeu 30928 pmtrcnel2 30936 cycpmconjslem1 30998 cycpmconjslem2 30999 reprpmtf1o 32176 subfacp1lem5 32717 ltrncoidN 37765 trlcoabs2N 38359 trlcoat 38360 trlcone 38365 cdlemg47 38373 tgrpgrplem 38386 tendoipl 38434 cdlemi2 38456 cdlemk2 38469 cdlemk4 38471 cdlemk8 38475 tendocnv 38658 dvhgrp 38744 cdlemn8 38841 dihopelvalcpre 38885 dssmap2d 41176 rngcinv 45073 rngcinvALTV 45085 ringcinv 45124 ringcinvALTV 45148 |
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