Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > setciso | Structured version Visualization version GIF version |
Description: An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
setcmon.c | β’ πΆ = (SetCatβπ) |
setcmon.u | β’ (π β π β π) |
setcmon.x | β’ (π β π β π) |
setcmon.y | β’ (π β π β π) |
setciso.n | β’ πΌ = (IsoβπΆ) |
Ref | Expression |
---|---|
setciso | β’ (π β (πΉ β (ππΌπ) β πΉ:πβ1-1-ontoβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . 4 β’ (BaseβπΆ) = (BaseβπΆ) | |
2 | eqid 2738 | . . . 4 β’ (InvβπΆ) = (InvβπΆ) | |
3 | setcmon.u | . . . . 5 β’ (π β π β π) | |
4 | setcmon.c | . . . . . 6 β’ πΆ = (SetCatβπ) | |
5 | 4 | setccat 17906 | . . . . 5 β’ (π β π β πΆ β Cat) |
6 | 3, 5 | syl 17 | . . . 4 β’ (π β πΆ β Cat) |
7 | setcmon.x | . . . . 5 β’ (π β π β π) | |
8 | 4, 3 | setcbas 17899 | . . . . 5 β’ (π β π = (BaseβπΆ)) |
9 | 7, 8 | eleqtrd 2841 | . . . 4 β’ (π β π β (BaseβπΆ)) |
10 | setcmon.y | . . . . 5 β’ (π β π β π) | |
11 | 10, 8 | eleqtrd 2841 | . . . 4 β’ (π β π β (BaseβπΆ)) |
12 | setciso.n | . . . 4 β’ πΌ = (IsoβπΆ) | |
13 | 1, 2, 6, 9, 11, 12 | isoval 17583 | . . 3 β’ (π β (ππΌπ) = dom (π(InvβπΆ)π)) |
14 | 13 | eleq2d 2824 | . 2 β’ (π β (πΉ β (ππΌπ) β πΉ β dom (π(InvβπΆ)π))) |
15 | 1, 2, 6, 9, 11 | invfun 17582 | . . . . 5 β’ (π β Fun (π(InvβπΆ)π)) |
16 | funfvbrb 6997 | . . . . 5 β’ (Fun (π(InvβπΆ)π) β (πΉ β dom (π(InvβπΆ)π) β πΉ(π(InvβπΆ)π)((π(InvβπΆ)π)βπΉ))) | |
17 | 15, 16 | syl 17 | . . . 4 β’ (π β (πΉ β dom (π(InvβπΆ)π) β πΉ(π(InvβπΆ)π)((π(InvβπΆ)π)βπΉ))) |
18 | 4, 3, 7, 10, 2 | setcinv 17911 | . . . . 5 β’ (π β (πΉ(π(InvβπΆ)π)((π(InvβπΆ)π)βπΉ) β (πΉ:πβ1-1-ontoβπ β§ ((π(InvβπΆ)π)βπΉ) = β‘πΉ))) |
19 | simpl 484 | . . . . 5 β’ ((πΉ:πβ1-1-ontoβπ β§ ((π(InvβπΆ)π)βπΉ) = β‘πΉ) β πΉ:πβ1-1-ontoβπ) | |
20 | 18, 19 | syl6bi 253 | . . . 4 β’ (π β (πΉ(π(InvβπΆ)π)((π(InvβπΆ)π)βπΉ) β πΉ:πβ1-1-ontoβπ)) |
21 | 17, 20 | sylbid 239 | . . 3 β’ (π β (πΉ β dom (π(InvβπΆ)π) β πΉ:πβ1-1-ontoβπ)) |
22 | eqid 2738 | . . . 4 β’ β‘πΉ = β‘πΉ | |
23 | 4, 3, 7, 10, 2 | setcinv 17911 | . . . . 5 β’ (π β (πΉ(π(InvβπΆ)π)β‘πΉ β (πΉ:πβ1-1-ontoβπ β§ β‘πΉ = β‘πΉ))) |
24 | funrel 6514 | . . . . . . 7 β’ (Fun (π(InvβπΆ)π) β Rel (π(InvβπΆ)π)) | |
25 | 15, 24 | syl 17 | . . . . . 6 β’ (π β Rel (π(InvβπΆ)π)) |
26 | releldm 5896 | . . . . . . 7 β’ ((Rel (π(InvβπΆ)π) β§ πΉ(π(InvβπΆ)π)β‘πΉ) β πΉ β dom (π(InvβπΆ)π)) | |
27 | 26 | ex 414 | . . . . . 6 β’ (Rel (π(InvβπΆ)π) β (πΉ(π(InvβπΆ)π)β‘πΉ β πΉ β dom (π(InvβπΆ)π))) |
28 | 25, 27 | syl 17 | . . . . 5 β’ (π β (πΉ(π(InvβπΆ)π)β‘πΉ β πΉ β dom (π(InvβπΆ)π))) |
29 | 23, 28 | sylbird 260 | . . . 4 β’ (π β ((πΉ:πβ1-1-ontoβπ β§ β‘πΉ = β‘πΉ) β πΉ β dom (π(InvβπΆ)π))) |
30 | 22, 29 | mpan2i 696 | . . 3 β’ (π β (πΉ:πβ1-1-ontoβπ β πΉ β dom (π(InvβπΆ)π))) |
31 | 21, 30 | impbid 211 | . 2 β’ (π β (πΉ β dom (π(InvβπΆ)π) β πΉ:πβ1-1-ontoβπ)) |
32 | 14, 31 | bitrd 279 | 1 β’ (π β (πΉ β (ππΌπ) β πΉ:πβ1-1-ontoβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5104 β‘ccnv 5630 dom cdm 5631 Rel wrel 5636 Fun wfun 6486 β1-1-ontoβwf1o 6491 βcfv 6492 (class class class)co 7350 Basecbs 17018 Catccat 17479 Invcinv 17563 Isociso 17564 SetCatcsetc 17896 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-fz 13354 df-struct 16954 df-slot 16989 df-ndx 17001 df-base 17019 df-hom 17092 df-cco 17093 df-cat 17483 df-cid 17484 df-sect 17565 df-inv 17566 df-iso 17567 df-setc 17897 |
This theorem is referenced by: yonffthlem 18106 |
Copyright terms: Public domain | W3C validator |