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Mirrors > Home > ILE Home > Th. List > f1ococnv1 | 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 5419 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
2 | dfrel2 5038 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | 1, 2 | sylib 121 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
4 | 3 | coeq2d 4750 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹)) |
5 | f1ocnv 5429 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
6 | f1ococnv2 5443 | . . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → (◡𝐹 ∘ ◡◡𝐹) = ( I ↾ 𝐴)) | |
7 | 5, 6 | syl 14 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = ( I ↾ 𝐴)) |
8 | 4, 7 | eqtr3d 2192 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 I cid 4250 ◡ccnv 4587 ↾ cres 4590 ∘ ccom 4592 Rel wrel 4593 –1-1-onto→wf1o 5171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 |
This theorem is referenced by: f1cocnv1 5446 f1ocnvfv1 5729 fcof1o 5741 mapen 6793 hashfacen 10718 |
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