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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 17426 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
2 | slotsbhcdif 17429 | . . . . 5 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) | |
3 | 2 | simp3i 1138 | . . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx) |
4 | 1, 3 | setsnid 17211 | . . 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 6915 | . . . . 5 ⊢ 𝐻 ∈ V |
7 | 6 | tposex 8275 | . . . 4 ⊢ tpos 𝐻 ∈ V |
8 | 1 | setsid 17210 | . . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉))) |
9 | 7, 8 | mpan2 689 | . . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉))) |
10 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
11 | eqid 2726 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
12 | oppchom.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
13 | 10, 5, 11, 12 | oppcval 17726 | . . . 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 6905 | . . 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 2792 | . 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
16 | tpos0 8271 | . . 3 ⊢ tpos ∅ = ∅ | |
17 | fvprc 6893 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅) | |
18 | 5, 17 | eqtrid 2778 | . . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅) |
19 | 18 | tposeqd 8244 | . . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅) |
20 | fvprc 6893 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
21 | 12, 20 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
22 | 21 | fveq2d 6905 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅)) |
23 | 1 | str0 17191 | . . . 4 ⊢ ∅ = (Hom ‘∅) |
24 | 22, 23 | eqtr4di 2784 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅) |
25 | 16, 19, 24 | 3eqtr4a 2792 | . 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 1534 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∅c0 4325 〈cop 4639 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 1st c1st 8001 2nd c2nd 8002 tpos ctpos 8240 sSet csts 17165 ndxcnx 17195 Basecbs 17213 Hom chom 17277 compcco 17278 oppCatcoppc 17724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-tpos 8241 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-hom 17290 df-cco 17291 df-oppc 17725 |
This theorem is referenced by: oppchom 17729 |
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