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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 6629 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1of 6617 | . . 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 6853 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ◡ccnv 5556 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 |
This theorem is referenced by: f1oiso2 7107 f1ocnvfv3 7154 uzrdglem 13328 uzrdgsuci 13331 fzennn 13339 cardfz 13341 fzfi 13343 iunmbl2 24160 f1otrg 26659 axcontlem10 26761 wlkiswwlks2lem5 27653 clwlkclwwlklem2a 27778 cnvbraval 29889 cnvbracl 29890 cycpmco2lem6 30775 cycpmco2 30777 mndpluscn 31171 ismtycnv 35082 rngoisocnv 35261 lautcnvclN 37226 lautcnvle 37227 lautcvr 37230 lautj 37231 lautm 37232 ltrncnvatb 37276 diacnvclN 38189 dihcnvcl 38409 dihlspsnat 38471 dihglblem6 38478 dochocss 38504 dochnoncon 38529 mapdcnvcl 38790 rmxyelxp 39516 isomuspgrlem1 43999 isomgrsym 44008 |
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