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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 set function are equinumerous. This variation of f1oeng 8967 does not require the Axiom of Replacement nor the Axiom of Power Sets. (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 6822 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | |
2 | 1 | spcegv 3588 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
3 | 2 | imp 408 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
4 | bren 8949 | . 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 class class class wbr 5149 –1-1-onto→wf1o 6543 ≈ cen 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-en 8940 |
This theorem is referenced by: f1oen2g 8964 unen 9046 domdifsn 9054 domunsncan 9072 sucdom2OLD 9082 sbthlem10 9092 domssex 9138 dif1enlemOLD 9157 pssnn 9168 f1oenfi 9182 f1oenfirn 9183 sbthfilem 9201 sucdom2 9206 phplem2OLD 9218 pssnnOLD 9265 f1finf1oOLD 9272 oien 9533 infdifsn 9652 fin4en1 10304 fin23lem21 10334 hashf1lem2 14417 odinf 19431 gsumval3lem2 19774 gsumval3 19775 hmphen2 23303 fnpreimac 31896 pibt2 36298 |
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