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Mirrors > Home > MPE Home > Th. List > oppciso | Structured version Visualization version GIF version |
Description: An isomorphism in the opposite category. See also remark 3.9 in [Adamek] p. 28. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcsect.b | ⊢ 𝐵 = (Base‘𝐶) |
oppcsect.o | ⊢ 𝑂 = (oppCat‘𝐶) |
oppcsect.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oppcsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
oppcsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
oppciso.s | ⊢ 𝐼 = (Iso‘𝐶) |
oppciso.t | ⊢ 𝐽 = (Iso‘𝑂) |
Ref | Expression |
---|---|
oppciso | ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcsect.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | oppcsect.o | . . . 4 ⊢ 𝑂 = (oppCat‘𝐶) | |
3 | oppcsect.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | oppcsect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | oppcsect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | eqid 2826 | . . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
7 | eqid 2826 | . . . 4 ⊢ (Inv‘𝑂) = (Inv‘𝑂) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oppcinv 16793 | . . 3 ⊢ (𝜑 → (𝑋(Inv‘𝑂)𝑌) = (𝑌(Inv‘𝐶)𝑋)) |
9 | 8 | dmeqd 5559 | . 2 ⊢ (𝜑 → dom (𝑋(Inv‘𝑂)𝑌) = dom (𝑌(Inv‘𝐶)𝑋)) |
10 | 2, 1 | oppcbas 16731 | . . 3 ⊢ 𝐵 = (Base‘𝑂) |
11 | 2 | oppccat 16735 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
12 | 3, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
13 | oppciso.t | . . 3 ⊢ 𝐽 = (Iso‘𝑂) | |
14 | 10, 7, 12, 4, 5, 13 | isoval 16778 | . 2 ⊢ (𝜑 → (𝑋𝐽𝑌) = dom (𝑋(Inv‘𝑂)𝑌)) |
15 | oppciso.s | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
16 | 1, 6, 3, 5, 4, 15 | isoval 16778 | . 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = dom (𝑌(Inv‘𝐶)𝑋)) |
17 | 9, 14, 16 | 3eqtr4d 2872 | 1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 dom cdm 5343 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 Catccat 16678 oppCatcoppc 16724 Invcinv 16758 Isociso 16759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-hom 16330 df-cco 16331 df-cat 16682 df-cid 16683 df-oppc 16725 df-sect 16760 df-inv 16761 df-iso 16762 |
This theorem is referenced by: (None) |
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