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Mirrors > Home > MPE Home > Th. List > f1ocnvfv1 | Structured version Visualization version 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 6863 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) | |
2 | 1 | fveq1d 6894 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ((◡𝐹 ∘ 𝐹)‘𝐶) = (( I ↾ 𝐴)‘𝐶)) |
3 | 2 | adantr 482 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (( I ↾ 𝐴)‘𝐶)) |
4 | f1of 6834 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
5 | fvco3 6991 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (◡𝐹‘(𝐹‘𝐶))) | |
6 | 4, 5 | sylan 581 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((◡𝐹 ∘ 𝐹)‘𝐶) = (◡𝐹‘(𝐹‘𝐶))) |
7 | fvresi 7171 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (( I ↾ 𝐴)‘𝐶) = 𝐶) | |
8 | 7 | adantl 483 | . 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (( I ↾ 𝐴)‘𝐶) = 𝐶) |
9 | 3, 6, 8 | 3eqtr3d 2781 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 I cid 5574 ◡ccnv 5676 ↾ cres 5679 ∘ ccom 5681 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 |
This theorem is referenced by: f1ocnvfv 7276 wemapwe 9692 fseqenlem2 10020 acndom 10046 isf34lem5 10373 axcc3 10433 pwfseqlem1 10653 hashdom 14339 fz1isolem 14422 cnrecnv 15112 sadcadd 16399 sadadd2 16401 invinv 17717 catcisolem 18060 mhmf1o 18682 srngnvl 20464 mdetleib2 22090 2ndcdisj 22960 cnheiborlem 24470 iunmbl2 25074 dvcnvlem 25493 eff1olem 26057 logef 26090 adjbdlnb 31337 cnvbrabra 31365 fsumiunle 32035 fzto1stinvn 32263 cycpmfv1 32272 cycpmfv2 32273 cycpmco2lem7 32291 madjusmdetlem1 32807 tpr2rico 32892 esumiun 33092 lautj 38964 lautm 38965 ldilcnv 38986 ltrneq2 39019 trlcnv 39036 diaocN 39996 dihcnvid1 40143 dochocss 40237 mapdcnvid1N 40525 sticksstones19 40981 nvocnvb 42173 isomushgr 46494 rngisom1 46718 |
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