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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 | ⊢ 𝐶 = (SetCat‘𝑈) |
setcmon.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
setcmon.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
setcmon.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
setciso.n | ⊢ 𝐼 = (Iso‘𝐶) |
Ref | Expression |
---|---|
setciso | ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | eqid 2818 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | setcmon.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | setcmon.c | . . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈) | |
5 | 4 | setccat 17333 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | setcmon.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
8 | 4, 3 | setcbas 17326 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) |
9 | 7, 8 | eleqtrd 2912 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
10 | setcmon.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
11 | 10, 8 | eleqtrd 2912 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
12 | setciso.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
13 | 1, 2, 6, 9, 11, 12 | isoval 17023 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
14 | 13 | eleq2d 2895 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
15 | 1, 2, 6, 9, 11 | invfun 17022 | . . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌)) |
16 | funfvbrb 6813 | . . . . 5 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹))) |
18 | 4, 3, 7, 10, 2 | setcinv 17338 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹))) |
19 | simpl 483 | . . . . 5 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹) → 𝐹:𝑋–1-1-onto→𝑌) | |
20 | 18, 19 | syl6bi 254 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹:𝑋–1-1-onto→𝑌)) |
21 | 17, 20 | sylbid 241 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹:𝑋–1-1-onto→𝑌)) |
22 | eqid 2818 | . . . 4 ⊢ ◡𝐹 = ◡𝐹 | |
23 | 4, 3, 7, 10, 2 | setcinv 17338 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ◡𝐹 = ◡𝐹))) |
24 | funrel 6365 | . . . . . . 7 ⊢ (Fun (𝑋(Inv‘𝐶)𝑌) → Rel (𝑋(Inv‘𝐶)𝑌)) | |
25 | 15, 24 | syl 17 | . . . . . 6 ⊢ (𝜑 → Rel (𝑋(Inv‘𝐶)𝑌)) |
26 | releldm 5807 | . . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)) | |
27 | 26 | ex 413 | . . . . . 6 ⊢ (Rel (𝑋(Inv‘𝐶)𝑌) → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
28 | 25, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
29 | 23, 28 | sylbird 261 | . . . 4 ⊢ (𝜑 → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ◡𝐹 = ◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
30 | 22, 29 | mpan2i 693 | . . 3 ⊢ (𝜑 → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
31 | 21, 30 | impbid 213 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) |
32 | 14, 31 | bitrd 280 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ◡ccnv 5547 dom cdm 5548 Rel wrel 5553 Fun wfun 6342 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Catccat 16923 Invcinv 17003 Isociso 17004 SetCatcsetc 17323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-hom 16577 df-cco 16578 df-cat 16927 df-cid 16928 df-sect 17005 df-inv 17006 df-iso 17007 df-setc 17324 |
This theorem is referenced by: yonffthlem 17520 |
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