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Mirrors > Home > MPE Home > Th. List > f1oen3g | Structured version Visualization version GIF version |
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8647 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
f1oen3g | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oeq1 6649 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | |
2 | 1 | spcegv 3512 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
3 | 2 | imp 410 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
4 | bren 8636 | . 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | |
5 | 3, 4 | sylibr 237 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1787 ∈ wcel 2110 class class class wbr 5053 –1-1-onto→wf1o 6379 ≈ cen 8623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-en 8627 |
This theorem is referenced by: f1oen2g 8645 unen 8723 domdifsn 8728 domunsncan 8745 sucdom2 8755 sbthlem10 8765 domssex 8807 phplem2 8826 dif1enlem 8838 pssnn 8846 f1oenfi 8859 f1oenfirn 8860 pssnnOLD 8895 f1finf1o 8902 oien 9154 infdifsn 9272 fin4en1 9923 fin23lem21 9953 hashf1lem2 14022 odinf 18954 gsumval3lem2 19291 gsumval3 19292 hmphen2 22696 fnpreimac 30728 pibt2 35325 |
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