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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 5490 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) | |
2 | 1 | fveq1d 5517 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ((◡𝐹 ∘ 𝐹)‘𝐶) = (( I ↾ 𝐴)‘𝐶)) |
3 | 2 | adantr 276 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (( I ↾ 𝐴)‘𝐶)) |
4 | f1of 5461 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | fvco3 5587 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (◡𝐹‘(𝐹‘𝐶))) | |
6 | 4, 5 | sylan 283 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (◡𝐹‘(𝐹‘𝐶))) |
7 | fvresi 5709 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐶) = 𝐶) | |
8 | 7 | adantl 277 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (( I ↾ 𝐴)‘𝐶) = 𝐶) |
9 | 3, 6, 8 | 3eqtr3d 2218 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 I cid 4288 ◡ccnv 4625 ↾ cres 4628 ∘ ccom 4630 ⟶wf 5212 –1-1-onto→wf1o 5215 ‘cfv 5216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 |
This theorem is referenced by: f1ocnvfv 5779 caseinl 7089 caseinr 7090 ctssdccl 7109 cc3 7266 iseqf1olemab 10486 cnrecnv 10914 fprodssdc 11593 ennnfonelemhf1o 12408 ennnfonelemex 12409 ennnfonelemrn 12414 ctinfomlemom 12422 ssnnctlemct 12441 mhmf1o 12815 isomninnlem 14660 iswomninnlem 14679 ismkvnnlem 14682 |
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