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Mirrors > Home > ILE Home > Th. List > f1ocnvfv1 | GIF version |
Description: The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Ref | Expression |
---|---|
f1ocnvfv1 | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ococnv1 5155 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) | |
2 | 1 | fveq1d 5180 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ((◡𝐹 ∘ 𝐹)‘𝐶) = (( I ↾ 𝐴)‘𝐶)) |
3 | 2 | adantr 261 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (( I ↾ 𝐴)‘𝐶)) |
4 | f1of 5126 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | fvco3 5244 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (◡𝐹‘(𝐹‘𝐶))) | |
6 | 4, 5 | sylan 267 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (◡𝐹‘(𝐹‘𝐶))) |
7 | fvresi 5356 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐶) = 𝐶) | |
8 | 7 | adantl 262 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (( I ↾ 𝐴)‘𝐶) = 𝐶) |
9 | 3, 6, 8 | 3eqtr3d 2080 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 I cid 4025 ◡ccnv 4344 ↾ cres 4347 ∘ ccom 4349 ⟶wf 4898 –1-1-onto→wf1o 4901 ‘cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 |
This theorem is referenced by: f1ocnvfv 5419 cnrecnv 9510 |
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