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 | ⊢ 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 2610 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | eqid 2610 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | setcmon.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | setcmon.c | . . . . . 6 ⊢ 𝐶 = (SetCat‘𝑈) | |
5 | 4 | setccat 16558 | . . . . 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 16551 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) |
9 | 7, 8 | eleqtrd 2690 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
10 | setcmon.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
11 | 10, 8 | eleqtrd 2690 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
12 | setciso.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
13 | 1, 2, 6, 9, 11, 12 | isoval 16248 | . . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌)) |
14 | 13 | eleq2d 2673 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
15 | 1, 2, 6, 9, 11 | invfun 16247 | . . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐶)𝑌)) |
16 | funfvbrb 6238 | . . . . 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 16563 | . . . . 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 472 | . . . . 5 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ((𝑋(Inv‘𝐶)𝑌)‘𝐹) = ◡𝐹) → 𝐹:𝑋–1-1-onto→𝑌) | |
20 | 18, 19 | syl6bi 242 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)((𝑋(Inv‘𝐶)𝑌)‘𝐹) → 𝐹:𝑋–1-1-onto→𝑌)) |
21 | 17, 20 | sylbid 229 | . . 3 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) → 𝐹:𝑋–1-1-onto→𝑌)) |
22 | eqid 2610 | . . . 4 ⊢ ◡𝐹 = ◡𝐹 | |
23 | 4, 3, 7, 10, 2 | setcinv 16563 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ◡𝐹 = ◡𝐹))) |
24 | funrel 5821 | . . . . . . 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 5279 | . . . . . . 7 ⊢ ((Rel (𝑋(Inv‘𝐶)𝑌) ∧ 𝐹(𝑋(Inv‘𝐶)𝑌)◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)) | |
27 | 26 | ex 449 | . . . . . 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 249 | . . . 4 ⊢ (𝜑 → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ◡𝐹 = ◡𝐹) → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
30 | 22, 29 | mpan2i 709 | . . 3 ⊢ (𝜑 → (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌))) |
31 | 21, 30 | impbid 201 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) |
32 | 14, 31 | bitrd 267 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ◡ccnv 5037 dom cdm 5038 Rel wrel 5043 Fun wfun 5798 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Catccat 16148 Invcinv 16228 Isociso 16229 SetCatcsetc 16548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-cat 16152 df-cid 16153 df-sect 16230 df-inv 16231 df-iso 16232 df-setc 16549 |
This theorem is referenced by: yonffthlem 16745 |
Copyright terms: Public domain | W3C validator |