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Mirrors > Home > MPE Home > Th. List > dfinito3 | Structured version Visualization version GIF version |
Description: An alternate definition of df-inito 17615 depending on df-termo 17616, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.) |
Ref | Expression |
---|---|
dfinito3 | ⊢ InitO = (TermO ∘ (oppCat ↾ Cat)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6775 | . . . 4 ⊢ (𝑐 ∈ Cat → ((oppCat ↾ Cat)‘𝑐) = (oppCat‘𝑐)) | |
2 | 1 | fveq2d 6760 | . . 3 ⊢ (𝑐 ∈ Cat → (TermO‘((oppCat ↾ Cat)‘𝑐)) = (TermO‘(oppCat‘𝑐))) |
3 | 2 | mpteq2ia 5173 | . 2 ⊢ (𝑐 ∈ Cat ↦ (TermO‘((oppCat ↾ Cat)‘𝑐))) = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐))) |
4 | termofn 17619 | . . . 4 ⊢ TermO Fn Cat | |
5 | dffn2 6586 | . . . 4 ⊢ (TermO Fn Cat ↔ TermO:Cat⟶V) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ TermO:Cat⟶V |
7 | oppccatf 17356 | . . 3 ⊢ (oppCat ↾ Cat):Cat⟶Cat | |
8 | fcompt 6987 | . . 3 ⊢ ((TermO:Cat⟶V ∧ (oppCat ↾ Cat):Cat⟶Cat) → (TermO ∘ (oppCat ↾ Cat)) = (𝑐 ∈ Cat ↦ (TermO‘((oppCat ↾ Cat)‘𝑐)))) | |
9 | 6, 7, 8 | mp2an 688 | . 2 ⊢ (TermO ∘ (oppCat ↾ Cat)) = (𝑐 ∈ Cat ↦ (TermO‘((oppCat ↾ Cat)‘𝑐))) |
10 | dfinito2 17634 | . 2 ⊢ InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐))) | |
11 | 3, 9, 10 | 3eqtr4ri 2777 | 1 ⊢ InitO = (TermO ∘ (oppCat ↾ Cat)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2108 Vcvv 3422 ↦ cmpt 5153 ↾ cres 5582 ∘ ccom 5584 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 Catccat 17290 oppCatcoppc 17337 InitOcinito 17612 TermOctermo 17613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-hom 16912 df-cco 16913 df-cat 17294 df-cid 17295 df-oppc 17338 df-inito 17615 df-termo 17616 |
This theorem is referenced by: (None) |
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