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Mirrors > Home > MPE Home > Th. List > f1ocnvdm | Structured version Visualization version GIF version |
Description: The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
Ref | Expression |
---|---|
f1ocnvdm | ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 6620 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1of 6608 | . . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵⟶𝐴) |
4 | 3 | ffvelrnda 6843 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ◡ccnv 5547 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: f1oiso2 7094 f1ocnvfv3 7141 uzrdglem 13313 uzrdgsuci 13316 fzennn 13324 cardfz 13326 fzfi 13328 iunmbl2 24085 f1otrg 26584 axcontlem10 26686 wlkiswwlks2lem5 27578 clwlkclwwlklem2a 27703 cnvbraval 29814 cnvbracl 29815 cycpmco2lem6 30700 cycpmco2 30702 mndpluscn 31068 ismtycnv 34961 rngoisocnv 35140 lautcnvclN 37104 lautcnvle 37105 lautcvr 37108 lautj 37109 lautm 37110 ltrncnvatb 37154 diacnvclN 38067 dihcnvcl 38287 dihlspsnat 38349 dihglblem6 38356 dochocss 38382 dochnoncon 38407 mapdcnvcl 38668 rmxyelxp 39387 isomuspgrlem1 43869 isomgrsym 43878 |
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