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Mirrors > Home > MPE Home > Th. List > bren | Structured version Visualization version GIF version |
Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) Extract breng 8826 as an intermediate result. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
bren | ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 8825 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | f1ofn 6781 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴) | |
3 | fndm 6601 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → dom 𝑓 = 𝐴) | |
4 | vex 3448 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
5 | 4 | dmex 7839 | . . . . . 6 ⊢ dom 𝑓 ∈ V |
6 | 3, 5 | eqeltrrdi 2848 | . . . . 5 ⊢ (𝑓 Fn 𝐴 → 𝐴 ∈ V) |
7 | 2, 6 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V) |
8 | f1ofo 6787 | . . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
9 | forn 6755 | . . . . . 6 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ran 𝑓 = 𝐵) |
11 | 4 | rnex 7840 | . . . . 5 ⊢ ran 𝑓 ∈ V |
12 | 10, 11 | eqeltrrdi 2848 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V) |
13 | 7, 12 | jca 513 | . . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
14 | 13 | exlimiv 1934 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
15 | breng 8826 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) | |
16 | 1, 14, 15 | pm5.21nii 380 | 1 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3444 class class class wbr 5104 dom cdm 5631 ran crn 5632 Fn wfn 6487 –onto→wfo 6490 –1-1-onto→wf1o 6491 ≈ cen 8814 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
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 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-en 8818 |
This theorem is referenced by: domen 8835 f1oen3g 8840 ener 8875 en0OLD 8892 en0ALT 8893 ensn1OLD 8896 en1OLD 8900 en2snOLD 8920 unen 8924 enfixsn 8959 canth2 9008 mapen 9019 ssenen 9029 rexdif1enOLD 9037 dif1en 9038 dif1enOLD 9040 ssfiALT 9052 ensymfib 9065 entrfil 9066 phplem2 9086 php3 9090 phplem4OLD 9098 php3OLD 9102 isinf 9138 isinfOLD 9139 domunfican 9198 fiint 9202 mapfien2 9279 unxpwdom2 9458 isinffi 9862 infxpenc2 9892 fseqen 9897 dfac8b 9901 infpwfien 9932 dfac12r 10016 infmap2 10088 cff1 10128 infpssr 10178 fin4en1 10179 enfin2i 10191 enfin1ai 10254 axcc3 10308 axcclem 10327 numth 10342 ttukey2g 10386 canthnum 10519 canthwe 10521 canthp1 10524 pwfseq 10534 tskuni 10653 gruen 10682 hasheqf1o 14177 hashfacen 14279 hashfacenOLD 14280 fz1f1o 15530 ruc 16060 cnso 16064 eulerth 16590 ablfaclem3 19795 lbslcic 21170 uvcendim 21176 indishmph 23071 ufldom 23235 ovolctb 24776 ovoliunlem3 24790 iunmbl2 24843 dyadmbl 24886 vitali 24899 cusgrfilem3 28191 padct 31418 f1ocnt 31487 volmeas 32591 eulerpart 32743 derangenlem 33526 mblfinlem1 36001 sticksstones4 40443 sticksstones20 40460 eldioph2lem1 40917 isnumbasgrplem1 41262 nnf1oxpnn 43135 sprsymrelen 45392 prproropen 45400 uspgrspren 45754 uspgrbisymrel 45756 1aryenef 46431 2aryenef 46442 rrx2xpreen 46505 |
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