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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 8914 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 6773 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | |
2 | 1 | spcegv 3555 | . . 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 8896 | . 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 5106 –1-1-onto→wf1o 6496 ≈ cen 8883 |
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 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-en 8887 |
This theorem is referenced by: f1oen2g 8911 unen 8993 domdifsn 9001 domunsncan 9019 sucdom2OLD 9029 sbthlem10 9039 domssex 9085 dif1enlemOLD 9104 pssnn 9115 f1oenfi 9129 f1oenfirn 9130 sbthfilem 9148 sucdom2 9153 phplem2OLD 9165 pssnnOLD 9212 f1finf1oOLD 9219 oien 9479 infdifsn 9598 fin4en1 10250 fin23lem21 10280 hashf1lem2 14361 odinf 19350 gsumval3lem2 19688 gsumval3 19689 hmphen2 23166 fnpreimac 31633 pibt2 35934 |
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