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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 6842 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1of 6830 | . . 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 | ffvelcdmda 7083 | 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 2106 ◡ccnv 5674 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 |
This theorem is referenced by: f1oiso2 7345 f1ocnvfv3 7400 dif1enlem 9152 dif1enlemOLD 9153 rexdif1en 9154 rexdif1enOLD 9155 dif1en 9156 dif1enOLD 9158 uzrdglem 13918 uzrdgsuci 13921 fzennn 13929 cardfz 13931 fzfi 13933 iunmbl2 25065 f1otrg 28111 axcontlem10 28220 wlkiswwlks2lem5 29116 clwlkclwwlklem2a 29240 cnvbraval 31350 cnvbracl 31351 cycpmco2lem6 32277 cycpmco2 32279 mndpluscn 32894 ismtycnv 36658 rngoisocnv 36837 lautcnvclN 38947 lautcnvle 38948 lautcvr 38951 lautj 38952 lautm 38953 ltrncnvatb 38997 diacnvclN 39910 dihcnvcl 40130 dihlspsnat 40192 dihglblem6 40199 dochocss 40225 dochnoncon 40250 mapdcnvcl 40511 rmxyelxp 41636 cantnfub 42056 isomuspgrlem1 46481 isomgrsym 46490 |
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