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| Mirrors > Home > MPE Home > Th. List > f1oen | Structured version Visualization version GIF version | ||
| Description: The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
| Ref | Expression |
|---|---|
| f1oen.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| f1oen | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | f1oen.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | f1oeng 9011 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 –1-1-onto→wf1o 6560 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-en 8986 |
| This theorem is referenced by: mapfien2 9449 infxpenlem 10053 dfac8alem 10069 dfac12lem2 10185 dfac12lem3 10186 r1om 10283 axcc2lem 10476 summolem3 15750 summolem2 15752 zsum 15754 prodmolem3 15969 prodmolem2 15971 zprod 15973 cpnnen 16265 eulerthlem2 16819 hashgcdeq 16827 4sqlem11 16993 gicen 19296 odhash 19592 odhash2 19593 sylow1lem2 19617 sylow2blem1 19638 znhash 21577 wlkswwlksen 29900 wlknwwlksnen 29909 eupthfi 30224 numclwwlk1lem2 30379 ballotlemfrc 34529 ballotlem8 34539 erdszelem10 35205 poimirlem4 37631 poimirlem26 37653 poimirlem27 37654 pwfi2en 43109 gricen 47894 grlicen 47977 thincciso2 49104 termcterm2 49146 aacllem 49320 |
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