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Mirrors > Home > ILE Home > Th. List > f1oeng | GIF version |
Description: The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
Ref | Expression |
---|---|
f1oeng | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ofo 5483 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | |
2 | focdmex 6134 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) | |
3 | 1, 2 | syl5 32 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)) |
4 | 3 | imp 124 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐵 ∈ V) |
5 | f1oen2g 6773 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
6 | 5 | 3com23 1211 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ V) → 𝐴 ≈ 𝐵) |
7 | 4, 6 | mpd3an3 1349 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 Vcvv 2752 class class class wbr 4018 –onto→wfo 5229 –1-1-onto→wf1o 5230 ≈ cen 6756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-en 6759 |
This theorem is referenced by: f1oen 6777 f1imaeng 6810 xpen 6863 fidifsnen 6888 dif1en 6897 f1ofi 6960 f1dmvrnfibi 6961 omp1eom 7112 endjusym 7113 eninl 7114 eninr 7115 summodclem2 11408 zsumdc 11410 prodmodclem2 11603 zproddc 11605 eulerthlemh 12249 ssnnctlemct 12465 conjsubgen 13178 pwf1oexmid 15134 |
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