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Mirrors > Home > ILE Home > Th. List > f1ocnvfv2 | GIF version |
Description: The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
Ref | Expression |
---|---|
f1ocnvfv2 | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ococnv2 5387 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) | |
2 | 1 | fveq1d 5416 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ((𝐹 ∘ ◡𝐹)‘𝐶) = (( I ↾ 𝐵)‘𝐶)) |
3 | 2 | adantr 274 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ((𝐹 ∘ ◡𝐹)‘𝐶) = (( I ↾ 𝐵)‘𝐶)) |
4 | f1ocnv 5373 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
5 | f1of 5360 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵⟶𝐴) |
7 | fvco3 5485 | . . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹 ∘ ◡𝐹)‘𝐶) = (𝐹‘(◡𝐹‘𝐶))) | |
8 | 6, 7 | sylan 281 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ((𝐹 ∘ ◡𝐹)‘𝐶) = (𝐹‘(◡𝐹‘𝐶))) |
9 | fvresi 5606 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (( I ↾ 𝐵)‘𝐶) = 𝐶) | |
10 | 9 | adantl 275 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (( I ↾ 𝐵)‘𝐶) = 𝐶) |
11 | 3, 8, 10 | 3eqtr3d 2178 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 I cid 4205 ◡ccnv 4533 ↾ cres 4536 ∘ ccom 4538 ⟶wf 5114 –1-1-onto→wf1o 5117 ‘cfv 5118 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: f1ocnvfvb 5674 isocnv 5705 f1oiso2 5721 ordiso2 6913 enomnilem 7003 frecuzrdglem 10177 frecuzrdgsuc 10180 frecuzrdgdomlem 10183 frecuzrdgsuctlem 10189 frecfzennn 10192 iseqf1olemkle 10250 iseqf1olemklt 10251 iseqf1olemnab 10254 seq3f1olemqsumkj 10264 hashfz1 10522 seq3coll 10578 summodclem3 11142 summodclem2a 11143 sqpweven 11842 2sqpwodd 11843 phimullem 11890 ennnfonelemkh 11914 ennnfonelemhf1o 11915 ennnfonelemex 11916 ennnfonelemnn0 11924 ctinfomlemom 11929 ctiunctlemfo 11941 isomninnlem 13214 |
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