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Mirrors > Home > ILE Home > Th. List > f1ocnv | GIF version |
Description: The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1ocnv | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5229 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | dfrel2 4997 | . . . . . 6 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | fneq1 5219 | . . . . . . 7 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
4 | 3 | biimprd 157 | . . . . . 6 ⊢ (◡◡𝐹 = 𝐹 → (𝐹 Fn 𝐴 → ◡◡𝐹 Fn 𝐴)) |
5 | 2, 4 | sylbi 120 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 Fn 𝐴 → ◡◡𝐹 Fn 𝐴)) |
6 | 1, 5 | mpcom 36 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ◡◡𝐹 Fn 𝐴) |
7 | 6 | anim2i 340 | . . 3 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐹 Fn 𝐴) → (◡𝐹 Fn 𝐵 ∧ ◡◡𝐹 Fn 𝐴)) |
8 | 7 | ancoms 266 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵) → (◡𝐹 Fn 𝐵 ∧ ◡◡𝐹 Fn 𝐴)) |
9 | dff1o4 5383 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | |
10 | dff1o4 5383 | . 2 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 ↔ (◡𝐹 Fn 𝐵 ∧ ◡◡𝐹 Fn 𝐴)) | |
11 | 8, 9, 10 | 3imtr4i 200 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ◡ccnv 4546 Rel wrel 4552 Fn wfn 5126 –1-1-onto→wf1o 5130 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 |
This theorem is referenced by: f1ocnvb 5389 f1orescnv 5391 f1imacnv 5392 f1cnv 5399 f1ococnv1 5404 f1oresrab 5593 f1ocnvfv2 5687 f1ocnvdm 5690 f1ocnvfvrneq 5691 fcof1o 5698 isocnv 5720 f1ofveu 5770 mapsnf1o3 6599 ener 6681 en0 6697 en1 6701 mapen 6748 ssenen 6753 preimaf1ofi 6847 ordiso2 6928 caseinl 6984 caseinr 6985 ctssdccl 7004 ctssdclemr 7005 enomnilem 7018 enmkvlem 7043 enwomnilem 7050 cc3 7100 fnn0nninf 10241 0tonninf 10243 1tonninf 10244 iseqf1olemkle 10288 iseqf1olemklt 10289 iseqf1olemqcl 10290 iseqf1olemnab 10292 iseqf1olemmo 10296 iseqf1olemqk 10298 seq3f1olemqsumkj 10302 seq3f1olemqsumk 10303 seq3f1olemstep 10305 hashfz1 10561 hashfacen 10611 seq3coll 10617 cnrecnv 10714 nnf1o 11177 summodclem3 11181 summodclem2a 11182 prodmodclem3 11376 prodmodclem2a 11377 sqpweven 11889 2sqpwodd 11890 phimullem 11937 xpnnen 11943 ennnfonelemjn 11951 ennnfonelemp1 11955 ennnfonelemhdmp1 11958 ennnfonelemss 11959 ennnfonelemkh 11961 ennnfonelemhf1o 11962 ennnfonelemex 11963 ennnfonelemf1 11967 ennnfonelemnn0 11971 ennnfonelemim 11973 ctinfomlemom 11976 ctiunctlemfo 11988 txhmeo 12527 dfrelog 12989 relogf1o 12990 012of 13363 exmidsbthrlem 13392 iswomninnlem 13417 |
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