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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 6062 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1of 6050 | . . 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 6267 | 1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ◡ccnv 5037 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: f1oiso2 6502 f1ocnvfv3 6545 uzrdglem 12618 uzrdgsuci 12621 fzennn 12629 cardfz 12631 fzfi 12633 iunmbl2 23132 f1otrg 25551 axcontlem10 25653 nbgraf1olem3 25972 nbgraf1olem5 25974 cusgrares 26001 constr3trllem1 26178 wlkiswwlk2lem5 26223 clwlkisclwwlklem2a 26313 cnvbraval 28353 cnvbracl 28354 mndpluscn 29300 ismtycnv 32771 rngoisocnv 32950 lautcnvclN 34392 lautcnvle 34393 lautcvr 34396 lautj 34397 lautm 34398 ltrncnvatb 34442 diacnvclN 35358 dihcnvcl 35578 dihlspsnat 35640 dihglblem6 35647 dochocss 35673 dochnoncon 35698 mapdcnvcl 35959 rmxyelxp 36495 1wlkiswwlks2lem5 41070 clwlkclwwlklem2a 41207 |
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