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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 2610 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2610 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2610 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | oppcbas.1 | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
6 | 2, 3, 4, 5 | oppcval 16196 | . . . . 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 6107 | . . . 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 15747 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
9 | 1re 9918 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
10 | 1nn 10908 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
11 | 4nn0 11188 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
12 | 1nn0 11185 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
13 | 1lt10 11557 | . . . . . . . . 9 ⊢ 1 < ;10 | |
14 | 10, 11, 12, 13 | declti 11422 | . . . . . . . 8 ⊢ 1 < ;14 |
15 | 9, 14 | ltneii 10029 | . . . . . . 7 ⊢ 1 ≠ ;14 |
16 | basendx 15751 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
17 | homndx 15897 | . . . . . . . 8 ⊢ (Hom ‘ndx) = ;14 | |
18 | 16, 17 | neeq12i 2848 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
19 | 15, 18 | mpbir 220 | . . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
20 | 8, 19 | setsnid 15743 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉)) |
21 | 5nn 11065 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ | |
22 | 4lt5 11077 | . . . . . . . . . 10 ⊢ 4 < 5 | |
23 | 12, 11, 21, 22 | declt 11406 | . . . . . . . . 9 ⊢ ;14 < ;15 |
24 | 4nn 11064 | . . . . . . . . . . . 12 ⊢ 4 ∈ ℕ | |
25 | 12, 24 | decnncl 11394 | . . . . . . . . . . 11 ⊢ ;14 ∈ ℕ |
26 | 25 | nnrei 10906 | . . . . . . . . . 10 ⊢ ;14 ∈ ℝ |
27 | 12, 21 | decnncl 11394 | . . . . . . . . . . 11 ⊢ ;15 ∈ ℕ |
28 | 27 | nnrei 10906 | . . . . . . . . . 10 ⊢ ;15 ∈ ℝ |
29 | 9, 26, 28 | lttri 10042 | . . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15) |
30 | 14, 23, 29 | mp2an 704 | . . . . . . . 8 ⊢ 1 < ;15 |
31 | 9, 30 | ltneii 10029 | . . . . . . 7 ⊢ 1 ≠ ;15 |
32 | ccondx 15899 | . . . . . . . 8 ⊢ (comp‘ndx) = ;15 | |
33 | 16, 32 | neeq12i 2848 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15) |
34 | 31, 33 | mpbir 220 | . . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx) |
35 | 8, 34 | setsnid 15743 | . . . . 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 2632 | . . . 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 2663 | . . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
38 | base0 15740 | . . . 4 ⊢ ∅ = (Base‘∅) | |
39 | fvprc 6097 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
40 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
41 | 5, 40 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
42 | 41 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅)) |
43 | 38, 39, 42 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
44 | 37, 43 | pm2.61i 175 | . 2 ⊢ (Base‘𝐶) = (Base‘𝑂) |
45 | 1, 44 | eqtri 2632 | 1 ⊢ 𝐵 = (Base‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 〈cop 4131 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 tpos ctpos 7238 1c1 9816 < clt 9953 4c4 10949 5c5 10950 ;cdc 11369 ndxcnx 15692 sSet csts 15693 Basecbs 15695 Hom chom 15779 compcco 15780 oppCatcoppc 16194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-hom 15793 df-cco 15794 df-oppc 16195 |
This theorem is referenced by: oppccatid 16202 oppchomf 16203 2oppcbas 16206 2oppccomf 16208 oppccomfpropd 16210 isepi 16223 epii 16226 oppcsect 16261 oppcsect2 16262 oppcinv 16263 oppciso 16264 sectepi 16267 episect 16268 funcoppc 16358 fulloppc 16405 fthoppc 16406 fthepi 16411 hofcl 16722 yon11 16727 yon12 16728 yon2 16729 oyon1cl 16734 yonedalem21 16736 yonedalem3a 16737 yonedalem4c 16740 yonedalem22 16741 yonedalem3b 16742 yonedalem3 16743 yonedainv 16744 yonffthlem 16745 |
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