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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.) (Proof shortened by AV, 14-Oct-2024.) |
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 17301 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
2 | slotsbhcdif 17304 | . . . . 5 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) | |
3 | 2 | simp3i 1142 | . . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx) |
4 | 1, 3 | setsnid 17089 | . . 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)))⟩)) |
5 | oppchom.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | 5 | fvexi 6860 | . . . . 5 ⊢ 𝐻 ∈ V |
7 | 6 | tposex 8195 | . . . 4 ⊢ tpos 𝐻 ∈ V |
8 | 1 | setsid 17088 | . . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))) |
9 | 7, 8 | mpan2 690 | . . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))) |
10 | eqid 2733 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
11 | eqid 2733 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
12 | oppchom.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
13 | 10, 5, 11, 12 | oppcval 17601 | . . . 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)))⟩)) |
14 | 13 | fveq2d 6850 | . . 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)))⟩))) |
15 | 4, 9, 14 | 3eqtr4a 2799 | . 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
16 | tpos0 8191 | . . 3 ⊢ tpos ∅ = ∅ | |
17 | fvprc 6838 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅) | |
18 | 5, 17 | eqtrid 2785 | . . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅) |
19 | 18 | tposeqd 8164 | . . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅) |
20 | fvprc 6838 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
21 | 12, 20 | eqtrid 2785 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
22 | 21 | fveq2d 6850 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅)) |
23 | 1 | str0 17069 | . . . 4 ⊢ ∅ = (Hom ‘∅) |
24 | 22, 23 | eqtr4di 2791 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅) |
25 | 16, 19, 24 | 3eqtr4a 2799 | . 2 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
26 | 15, 25 | pm2.61i 182 | 1 ⊢ tpos 𝐻 = (Hom ‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ∅c0 4286 ⟨cop 4596 × cxp 5635 ‘cfv 6500 (class class class)co 7361 ∈ cmpo 7363 1st c1st 7923 2nd c2nd 7924 tpos ctpos 8160 sSet csts 17043 ndxcnx 17073 Basecbs 17091 Hom chom 17152 compcco 17153 oppCatcoppc 17599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-hom 17165 df-cco 17166 df-oppc 17600 |
This theorem is referenced by: oppchom 17604 |
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