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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 8986 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
3 | 1, 2 | mpan 689 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3470 class class class wbr 5143 –1-1-onto→wf1o 6542 ≈ cen 8955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-en 8959 |
This theorem is referenced by: mapfien2 9427 infxpenlem 10031 dfac8alem 10047 dfac12lem2 10162 dfac12lem3 10163 r1om 10262 axcc2lem 10454 summolem3 15687 summolem2 15689 zsum 15691 prodmolem3 15904 prodmolem2 15906 zprod 15908 cpnnen 16200 eulerthlem2 16745 hashgcdeq 16752 4sqlem11 16918 gicen 19226 odhash 19523 odhash2 19524 sylow1lem2 19548 sylow2blem1 19569 znhash 21486 wlkswwlksen 29685 wlknwwlksnen 29694 eupthfi 30009 numclwwlk1lem2 30164 ballotlemfrc 34141 ballotlem8 34151 erdszelem10 34805 poimirlem4 37092 poimirlem26 37114 poimirlem27 37115 pwfi2en 42512 gricen 47182 aacllem 48225 |
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