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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 16690 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
2 | 1nn0 11916 | . . . . . . . 8 ⊢ 1 ∈ ℕ0 | |
3 | 4nn 11723 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
4 | 2, 3 | decnncl 12121 | . . . . . . 7 ⊢ ;14 ∈ ℕ |
5 | 4 | nnrei 11649 | . . . . . 6 ⊢ ;14 ∈ ℝ |
6 | 4nn0 11919 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
7 | 5nn 11726 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
8 | 4lt5 11817 | . . . . . . 7 ⊢ 4 < 5 | |
9 | 2, 6, 7, 8 | declt 12129 | . . . . . 6 ⊢ ;14 < ;15 |
10 | 5, 9 | ltneii 10755 | . . . . 5 ⊢ ;14 ≠ ;15 |
11 | homndx 16689 | . . . . . 6 ⊢ (Hom ‘ndx) = ;14 | |
12 | ccondx 16691 | . . . . . 6 ⊢ (comp‘ndx) = ;15 | |
13 | 11, 12 | neeq12i 3084 | . . . . 5 ⊢ ((Hom ‘ndx) ≠ (comp‘ndx) ↔ ;14 ≠ ;15) |
14 | 10, 13 | mpbir 233 | . . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx) |
15 | 1, 14 | setsnid 16541 | . . 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 | 16 | fvexi 6686 | . . . . 5 ⊢ 𝐻 ∈ V |
18 | 17 | tposex 7928 | . . . 4 ⊢ tpos 𝐻 ∈ V |
19 | 1 | setsid 16540 | . . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉))) |
20 | 18, 19 | mpan2 689 | . . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉))) |
21 | eqid 2823 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
22 | eqid 2823 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
23 | oppchom.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
24 | 21, 16, 22, 23 | oppcval 16985 | . . . 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)))〉)) |
25 | 24 | fveq2d 6676 | . . 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)))〉))) |
26 | 15, 20, 25 | 3eqtr4a 2884 | . 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
27 | tpos0 7924 | . . 3 ⊢ tpos ∅ = ∅ | |
28 | fvprc 6665 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅) | |
29 | 16, 28 | syl5eq 2870 | . . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅) |
30 | 29 | tposeqd 7897 | . . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅) |
31 | fvprc 6665 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
32 | 23, 31 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
33 | 32 | fveq2d 6676 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅)) |
34 | df-hom 16591 | . . . . 5 ⊢ Hom = Slot ;14 | |
35 | 34 | str0 16537 | . . . 4 ⊢ ∅ = (Hom ‘∅) |
36 | 33, 35 | syl6eqr 2876 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅) |
37 | 27, 30, 36 | 3eqtr4a 2884 | . 2 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂)) |
38 | 26, 37 | pm2.61i 184 | 1 ⊢ tpos 𝐻 = (Hom ‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 〈cop 4575 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1st c1st 7689 2nd c2nd 7690 tpos ctpos 7893 1c1 10540 4c4 11697 5c5 11698 ;cdc 12101 ndxcnx 16482 sSet csts 16483 Basecbs 16485 Hom chom 16578 compcco 16579 oppCatcoppc 16983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 df-ndx 16488 df-slot 16489 df-sets 16492 df-hom 16591 df-cco 16592 df-oppc 16984 |
This theorem is referenced by: oppchom 16987 |
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