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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 9009 | . 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 2105 Vcvv 3477 class class class wbr 5147 –1-1-onto→wf1o 6561 ≈ cen 8980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-en 8984 |
This theorem is referenced by: mapfien2 9446 infxpenlem 10050 dfac8alem 10066 dfac12lem2 10182 dfac12lem3 10183 r1om 10280 axcc2lem 10473 summolem3 15746 summolem2 15748 zsum 15750 prodmolem3 15965 prodmolem2 15967 zprod 15969 cpnnen 16261 eulerthlem2 16815 hashgcdeq 16822 4sqlem11 16988 gicen 19308 odhash 19606 odhash2 19607 sylow1lem2 19631 sylow2blem1 19652 znhash 21594 wlkswwlksen 29909 wlknwwlksnen 29918 eupthfi 30233 numclwwlk1lem2 30388 ballotlemfrc 34507 ballotlem8 34517 erdszelem10 35184 poimirlem4 37610 poimirlem26 37632 poimirlem27 37633 pwfi2en 43085 gricen 47831 grlicen 47912 aacllem 49031 |
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