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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 2733 | . . . 4 β’ (BaseβπΆ) = (BaseβπΆ) | |
2 | eqid 2733 | . . . 4 β’ (InvβπΆ) = (InvβπΆ) | |
3 | setcmon.u | . . . . 5 β’ (π β π β π) | |
4 | setcmon.c | . . . . . 6 β’ πΆ = (SetCatβπ) | |
5 | 4 | setccat 18035 | . . . . 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 18028 | . . . . 5 β’ (π β π = (BaseβπΆ)) |
9 | 7, 8 | eleqtrd 2836 | . . . 4 β’ (π β π β (BaseβπΆ)) |
10 | setcmon.y | . . . . 5 β’ (π β π β π) | |
11 | 10, 8 | eleqtrd 2836 | . . . 4 β’ (π β π β (BaseβπΆ)) |
12 | setciso.n | . . . 4 β’ πΌ = (IsoβπΆ) | |
13 | 1, 2, 6, 9, 11, 12 | isoval 17712 | . . 3 β’ (π β (ππΌπ) = dom (π(InvβπΆ)π)) |
14 | 13 | eleq2d 2820 | . 2 β’ (π β (πΉ β (ππΌπ) β πΉ β dom (π(InvβπΆ)π))) |
15 | 1, 2, 6, 9, 11 | invfun 17711 | . . . . 5 β’ (π β Fun (π(InvβπΆ)π)) |
16 | funfvbrb 7053 | . . . . 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 18040 | . . . . 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 2733 | . . . 4 β’ β‘πΉ = β‘πΉ | |
23 | 4, 3, 7, 10, 2 | setcinv 18040 | . . . . 5 β’ (π β (πΉ(π(InvβπΆ)π)β‘πΉ β (πΉ:πβ1-1-ontoβπ β§ β‘πΉ = β‘πΉ))) |
24 | funrel 6566 | . . . . . . 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 5944 | . . . . . . 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 5149 β‘ccnv 5676 dom cdm 5677 Rel wrel 5682 Fun wfun 6538 β1-1-ontoβwf1o 6543 βcfv 6544 (class class class)co 7409 Basecbs 17144 Catccat 17608 Invcinv 17692 Isociso 17693 SetCatcsetc 18025 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-hom 17221 df-cco 17222 df-cat 17612 df-cid 17613 df-sect 17694 df-inv 17695 df-iso 17696 df-setc 18026 |
This theorem is referenced by: yonffthlem 18235 |
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