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Mirrors > Home > MPE Home > Th. List > oppcbas | Structured version Visualization version GIF version |
Description: Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
oppcbas.1 | ⊢ 𝑂 = (oppCat‘𝐶) |
oppcbas.2 | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
oppcbas | ⊢ 𝐵 = (Base‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcbas.2 | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2651 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2651 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2651 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | oppcbas.1 | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
6 | 2, 3, 4, 5 | oppcval 16420 | . . . . 5 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉)) |
7 | 6 | fveq2d 6233 | . . . 4 ⊢ (𝐶 ∈ V → (Base‘𝑂) = (Base‘((𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉))) |
8 | baseid 15966 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
9 | 1re 10077 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
10 | 1nn 11069 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
11 | 4nn0 11349 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
12 | 1nn0 11346 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
13 | 1lt10 11719 | . . . . . . . . 9 ⊢ 1 < ;10 | |
14 | 10, 11, 12, 13 | declti 11584 | . . . . . . . 8 ⊢ 1 < ;14 |
15 | 9, 14 | ltneii 10188 | . . . . . . 7 ⊢ 1 ≠ ;14 |
16 | basendx 15970 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
17 | homndx 16121 | . . . . . . . 8 ⊢ (Hom ‘ndx) = ;14 | |
18 | 16, 17 | neeq12i 2889 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
19 | 15, 18 | mpbir 221 | . . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
20 | 8, 19 | setsnid 15962 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉)) |
21 | 5nn 11226 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ | |
22 | 4lt5 11238 | . . . . . . . . . 10 ⊢ 4 < 5 | |
23 | 12, 11, 21, 22 | declt 11568 | . . . . . . . . 9 ⊢ ;14 < ;15 |
24 | 4nn 11225 | . . . . . . . . . . . 12 ⊢ 4 ∈ ℕ | |
25 | 12, 24 | decnncl 11556 | . . . . . . . . . . 11 ⊢ ;14 ∈ ℕ |
26 | 25 | nnrei 11067 | . . . . . . . . . 10 ⊢ ;14 ∈ ℝ |
27 | 12, 21 | decnncl 11556 | . . . . . . . . . . 11 ⊢ ;15 ∈ ℕ |
28 | 27 | nnrei 11067 | . . . . . . . . . 10 ⊢ ;15 ∈ ℝ |
29 | 9, 26, 28 | lttri 10201 | . . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15) |
30 | 14, 23, 29 | mp2an 708 | . . . . . . . 8 ⊢ 1 < ;15 |
31 | 9, 30 | ltneii 10188 | . . . . . . 7 ⊢ 1 ≠ ;15 |
32 | ccondx 16123 | . . . . . . . 8 ⊢ (comp‘ndx) = ;15 | |
33 | 16, 32 | neeq12i 2889 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15) |
34 | 31, 33 | mpbir 221 | . . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx) |
35 | 8, 34 | setsnid 15962 | . . . . 5 ⊢ (Base‘(𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉)) = (Base‘((𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉)) |
36 | 20, 35 | eqtri 2673 | . . . 4 ⊢ (Base‘𝐶) = (Base‘((𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝐶)(1st ‘𝑢)))〉)) |
37 | 7, 36 | syl6reqr 2704 | . . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
38 | base0 15959 | . . . 4 ⊢ ∅ = (Base‘∅) | |
39 | fvprc 6223 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
40 | fvprc 6223 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
41 | 5, 40 | syl5eq 2697 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
42 | 41 | fveq2d 6233 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅)) |
43 | 38, 39, 42 | 3eqtr4a 2711 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
44 | 37, 43 | pm2.61i 176 | . 2 ⊢ (Base‘𝐶) = (Base‘𝑂) |
45 | 1, 44 | eqtri 2673 | 1 ⊢ 𝐵 = (Base‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 Vcvv 3231 ∅c0 3948 〈cop 4216 class class class wbr 4685 × cxp 5141 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 1st c1st 7208 2nd c2nd 7209 tpos ctpos 7396 1c1 9975 < clt 10112 4c4 11110 5c5 11111 ;cdc 11531 ndxcnx 15901 sSet csts 15902 Basecbs 15904 Hom chom 15999 compcco 16000 oppCatcoppc 16418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-hom 16013 df-cco 16014 df-oppc 16419 |
This theorem is referenced by: oppccatid 16426 oppchomf 16427 2oppcbas 16430 2oppccomf 16432 oppccomfpropd 16434 isepi 16447 epii 16450 oppcsect 16485 oppcsect2 16486 oppcinv 16487 oppciso 16488 sectepi 16491 episect 16492 funcoppc 16582 fulloppc 16629 fthoppc 16630 fthepi 16635 hofcl 16946 yon11 16951 yon12 16952 yon2 16953 oyon1cl 16958 yonedalem21 16960 yonedalem3a 16961 yonedalem4c 16964 yonedalem22 16965 yonedalem3b 16966 yonedalem3 16967 yonedainv 16968 yonffthlem 16969 |
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