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Mirrors > Home > MPE Home > Th. List > dfinito2 | Structured version Visualization version GIF version |
Description: An initial object is a terminal object in the opposite category. An alternate definition of df-inito 17710 depending on df-termo 17711. (Contributed by Zhi Wang, 29-Aug-2024.) |
Ref | Expression |
---|---|
dfinito2 | ⊢ InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inito 17710 | . 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) | |
2 | eqid 2740 | . . . . . 6 ⊢ (oppCat‘𝑐) = (oppCat‘𝑐) | |
3 | 2 | oppccat 17444 | . . . . 5 ⊢ (𝑐 ∈ Cat → (oppCat‘𝑐) ∈ Cat) |
4 | eqid 2740 | . . . . . 6 ⊢ (Base‘𝑐) = (Base‘𝑐) | |
5 | 2, 4 | oppcbas 17439 | . . . . 5 ⊢ (Base‘𝑐) = (Base‘(oppCat‘𝑐)) |
6 | eqid 2740 | . . . . 5 ⊢ (Hom ‘(oppCat‘𝑐)) = (Hom ‘(oppCat‘𝑐)) | |
7 | 3, 5, 6 | termoval 17720 | . . . 4 ⊢ (𝑐 ∈ Cat → (TermO‘(oppCat‘𝑐)) = {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘(oppCat‘𝑐))𝑎)}) |
8 | eqid 2740 | . . . . . . . . 9 ⊢ (Hom ‘𝑐) = (Hom ‘𝑐) | |
9 | 8, 2 | oppchom 17436 | . . . . . . . 8 ⊢ (𝑏(Hom ‘(oppCat‘𝑐))𝑎) = (𝑎(Hom ‘𝑐)𝑏) |
10 | 9 | eleq2i 2832 | . . . . . . 7 ⊢ (ℎ ∈ (𝑏(Hom ‘(oppCat‘𝑐))𝑎) ↔ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)) |
11 | 10 | eubii 2587 | . . . . . 6 ⊢ (∃!ℎ ℎ ∈ (𝑏(Hom ‘(oppCat‘𝑐))𝑎) ↔ ∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)) |
12 | 11 | ralbii 3093 | . . . . 5 ⊢ (∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘(oppCat‘𝑐))𝑎) ↔ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)) |
13 | 12 | rabbii 3406 | . . . 4 ⊢ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘(oppCat‘𝑐))𝑎)} = {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)} |
14 | 7, 13 | eqtrdi 2796 | . . 3 ⊢ (𝑐 ∈ Cat → (TermO‘(oppCat‘𝑐)) = {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) |
15 | 14 | mpteq2ia 5182 | . 2 ⊢ (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐))) = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) |
16 | 1, 15 | eqtr4i 2771 | 1 ⊢ InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ∃!weu 2570 ∀wral 3066 {crab 3070 ↦ cmpt 5162 ‘cfv 6432 (class class class)co 7272 Basecbs 16923 Hom chom 16984 Catccat 17384 oppCatcoppc 17431 InitOcinito 17707 TermOctermo 17708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-tpos 8034 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-hom 16997 df-cco 16998 df-cat 17388 df-cid 17389 df-oppc 17432 df-inito 17710 df-termo 17711 |
This theorem is referenced by: dfinito3 17731 |
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