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Mirrors > Home > MPE Home > Th. List > oppchomfval | Structured version Visualization version GIF version |
Description: Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
oppchom.h | ⊢ 𝐻 = (Hom ‘𝐶) |
oppchom.o | ⊢ 𝑂 = (oppCat‘𝐶) |
Ref | Expression |
---|---|
oppchomfval | ⊢ tpos 𝐻 = (Hom ‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | homid 16284 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
2 | 1nn0 11511 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
3 | 4nn 11390 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
4 | 2, 3 | decnncl 11721 | . . . . . . 7 ⊢ ;14 ∈ ℕ |
5 | 4 | nnrei 11232 | . . . . . 6 ⊢ ;14 ∈ ℝ |
6 | 4nn0 11514 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
7 | 5nn 11391 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
8 | 4lt5 11403 | . . . . . . 7 ⊢ 4 < 5 | |
9 | 2, 6, 7, 8 | declt 11733 | . . . . . 6 ⊢ ;14 < ;15 |
10 | 5, 9 | ltneii 10353 | . . . . 5 ⊢ ;14 ≠ ;15 |
11 | homndx 16283 | . . . . . 6 ⊢ (Hom ‘ndx) = ;14 | |
12 | ccondx 16285 | . . . . . 6 ⊢ (comp‘ndx) = ;15 | |
13 | 11, 12 | neeq12i 3009 | . . . . 5 ⊢ ((Hom ‘ndx) ≠ (comp‘ndx) ↔ ;14 ≠ ;15) |
14 | 10, 13 | mpbir 221 | . . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx) |
15 | 1, 14 | setsnid 16123 | . . 3 ⊢ (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉)) = (Hom ‘((𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉)) |
16 | oppchom.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
17 | fvex 6343 | . . . . . 6 ⊢ (Hom ‘𝐶) ∈ V | |
18 | 16, 17 | eqeltri 2846 | . . . . 5 ⊢ 𝐻 ∈ V |
19 | 18 | tposex 7539 | . . . 4 ⊢ tpos 𝐻 ∈ V |
20 | 1 | setsid 16122 | . . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉))) |
21 | 19, 20 | mpan2 665 | . . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉))) |
22 | eqid 2771 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
23 | eqid 2771 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
24 | oppchom.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
25 | 22, 16, 23, 24 | oppcval 16581 | . . . 4 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉)) |
26 | 25 | fveq2d 6337 | . . 3 ⊢ (𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘((𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉))) |
27 | 15, 21, 26 | 3eqtr4a 2831 | . 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
28 | tpos0 7535 | . . 3 ⊢ tpos ∅ = ∅ | |
29 | fvprc 6327 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅) | |
30 | 16, 29 | syl5eq 2817 | . . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅) |
31 | 30 | tposeqd 7508 | . . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅) |
32 | fvprc 6327 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
33 | 24, 32 | syl5eq 2817 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
34 | 33 | fveq2d 6337 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅)) |
35 | df-hom 16175 | . . . . 5 ⊢ Hom = Slot ;14 | |
36 | 35 | str0 16119 | . . . 4 ⊢ ∅ = (Hom ‘∅) |
37 | 34, 36 | syl6eqr 2823 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅) |
38 | 28, 31, 37 | 3eqtr4a 2831 | . 2 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
39 | 27, 38 | pm2.61i 176 | 1 ⊢ tpos 𝐻 = (Hom ‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 Vcvv 3351 ∅c0 4064 〈cop 4323 × cxp 5248 ‘cfv 6032 (class class class)co 6794 ↦ cmpt2 6796 1st c1st 7314 2nd c2nd 7315 tpos ctpos 7504 1c1 10140 4c4 11275 5c5 11276 ;cdc 11696 ndxcnx 16062 sSet csts 16063 Basecbs 16065 Hom chom 16161 compcco 16162 oppCatcoppc 16579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-tpos 7505 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-nn 11224 df-2 11282 df-3 11283 df-4 11284 df-5 11285 df-6 11286 df-7 11287 df-8 11288 df-9 11289 df-n0 11496 df-dec 11697 df-ndx 16068 df-slot 16069 df-sets 16072 df-hom 16175 df-cco 16176 df-oppc 16580 |
This theorem is referenced by: oppchom 16583 |
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