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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 8531 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 –1-1-onto→wf1o 6357 ≈ cen 8509 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-en 8513 |
This theorem is referenced by: mapfien2 8875 infxpenlem 9442 dfac8alem 9458 dfac12lem2 9573 dfac12lem3 9574 r1om 9669 axcc2lem 9861 summolem3 15074 summolem2 15076 zsum 15078 prodmolem3 15290 prodmolem2 15292 zprod 15294 cpnnen 15585 eulerthlem2 16122 hashgcdeq 16129 4sqlem11 16294 gicen 18420 odhash 18702 odhash2 18703 sylow1lem2 18727 sylow2blem1 18748 znhash 20708 wlkswwlksen 27661 wlknwwlksnen 27670 eupthfi 27987 numclwwlk1lem2 28142 ballotlemfrc 31788 ballotlem8 31798 erdszelem10 32451 poimirlem4 34900 poimirlem26 34922 poimirlem27 34923 pwfi2en 39703 aacllem 44909 |
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