Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > oppccatf | Structured version Visualization version GIF version |
Description: oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.) |
Ref | Expression |
---|---|
oppccatf | ⊢ (oppCat ↾ Cat):Cat⟶Cat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oppc 17054 | . . . 4 ⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝑓)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉)) | |
2 | 1 | funmpt2 6379 | . . 3 ⊢ Fun oppCat |
3 | ffvresb 6885 | . . 3 ⊢ (Fun oppCat → ((oppCat ↾ Cat):Cat⟶Cat ↔ ∀𝑐 ∈ Cat (𝑐 ∈ dom oppCat ∧ (oppCat‘𝑐) ∈ Cat))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((oppCat ↾ Cat):Cat⟶Cat ↔ ∀𝑐 ∈ Cat (𝑐 ∈ dom oppCat ∧ (oppCat‘𝑐) ∈ Cat)) |
5 | elex 3428 | . . . 4 ⊢ (𝑐 ∈ Cat → 𝑐 ∈ V) | |
6 | ovex 7189 | . . . . 5 ⊢ ((𝑓 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝑓)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉) ∈ V | |
7 | 6, 1 | dmmpti 6480 | . . . 4 ⊢ dom oppCat = V |
8 | 5, 7 | eleqtrrdi 2863 | . . 3 ⊢ (𝑐 ∈ Cat → 𝑐 ∈ dom oppCat) |
9 | eqid 2758 | . . . 4 ⊢ (oppCat‘𝑐) = (oppCat‘𝑐) | |
10 | 9 | oppccat 17064 | . . 3 ⊢ (𝑐 ∈ Cat → (oppCat‘𝑐) ∈ Cat) |
11 | 8, 10 | jca 515 | . 2 ⊢ (𝑐 ∈ Cat → (𝑐 ∈ dom oppCat ∧ (oppCat‘𝑐) ∈ Cat)) |
12 | 4, 11 | mprgbir 3085 | 1 ⊢ (oppCat ↾ Cat):Cat⟶Cat |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∀wral 3070 Vcvv 3409 〈cop 4531 × cxp 5526 dom cdm 5528 ↾ cres 5530 Fun wfun 6334 ⟶wf 6336 ‘cfv 6340 (class class class)co 7156 ∈ cmpo 7158 1st c1st 7697 2nd c2nd 7698 tpos ctpos 7907 ndxcnx 16552 sSet csts 16553 Basecbs 16555 Hom chom 16648 compcco 16649 Catccat 17007 oppCatcoppc 17053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-tpos 7908 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-er 8305 df-en 8541 df-dom 8542 df-sdom 8543 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-ndx 16558 df-slot 16559 df-base 16561 df-sets 16562 df-hom 16661 df-cco 16662 df-cat 17011 df-cid 17012 df-oppc 17054 |
This theorem is referenced by: dfinito3 17345 dftermo3 17346 |
Copyright terms: Public domain | W3C validator |