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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 2823 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | eqid 2823 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | eqid 2823 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
5 | oppcbas.1 | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
6 | 2, 3, 4, 5 | oppcval 16985 | . . . . 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 6676 | . . . 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 16545 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
9 | 1re 10643 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
10 | 1nn 11651 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
11 | 4nn0 11919 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
12 | 1nn0 11916 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
13 | 1lt10 12240 | . . . . . . . . 9 ⊢ 1 < ;10 | |
14 | 10, 11, 12, 13 | declti 12139 | . . . . . . . 8 ⊢ 1 < ;14 |
15 | 9, 14 | ltneii 10755 | . . . . . . 7 ⊢ 1 ≠ ;14 |
16 | basendx 16549 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
17 | homndx 16689 | . . . . . . . 8 ⊢ (Hom ‘ndx) = ;14 | |
18 | 16, 17 | neeq12i 3084 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
19 | 15, 18 | mpbir 233 | . . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
20 | 8, 19 | setsnid 16541 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉)) |
21 | 5nn 11726 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ | |
22 | 4lt5 11817 | . . . . . . . . . 10 ⊢ 4 < 5 | |
23 | 12, 11, 21, 22 | declt 12129 | . . . . . . . . 9 ⊢ ;14 < ;15 |
24 | 4nn 11723 | . . . . . . . . . . . 12 ⊢ 4 ∈ ℕ | |
25 | 12, 24 | decnncl 12121 | . . . . . . . . . . 11 ⊢ ;14 ∈ ℕ |
26 | 25 | nnrei 11649 | . . . . . . . . . 10 ⊢ ;14 ∈ ℝ |
27 | 12, 21 | decnncl 12121 | . . . . . . . . . . 11 ⊢ ;15 ∈ ℕ |
28 | 27 | nnrei 11649 | . . . . . . . . . 10 ⊢ ;15 ∈ ℝ |
29 | 9, 26, 28 | lttri 10768 | . . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15) |
30 | 14, 23, 29 | mp2an 690 | . . . . . . . 8 ⊢ 1 < ;15 |
31 | 9, 30 | ltneii 10755 | . . . . . . 7 ⊢ 1 ≠ ;15 |
32 | ccondx 16691 | . . . . . . . 8 ⊢ (comp‘ndx) = ;15 | |
33 | 16, 32 | neeq12i 3084 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15) |
34 | 31, 33 | mpbir 233 | . . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx) |
35 | 8, 34 | setsnid 16541 | . . . . 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 2846 | . . . 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 2877 | . . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
38 | base0 16538 | . . . 4 ⊢ ∅ = (Base‘∅) | |
39 | fvprc 6665 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
40 | fvprc 6665 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
41 | 5, 40 | syl5eq 2870 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
42 | 41 | fveq2d 6676 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅)) |
43 | 38, 39, 42 | 3eqtr4a 2884 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
44 | 37, 43 | pm2.61i 184 | . 2 ⊢ (Base‘𝐶) = (Base‘𝑂) |
45 | 1, 44 | eqtri 2846 | 1 ⊢ 𝐵 = (Base‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 〈cop 4575 class class class wbr 5068 × cxp 5555 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1st c1st 7689 2nd c2nd 7690 tpos ctpos 7893 1c1 10540 < clt 10677 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-z 11985 df-dec 12102 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-hom 16591 df-cco 16592 df-oppc 16984 |
This theorem is referenced by: oppccatid 16991 oppchomf 16992 2oppcbas 16995 2oppccomf 16997 oppccomfpropd 16999 isepi 17012 epii 17015 oppcsect 17050 oppcsect2 17051 oppcinv 17052 oppciso 17053 sectepi 17056 episect 17057 funcoppc 17147 fulloppc 17194 fthoppc 17195 fthepi 17200 hofcl 17511 yon11 17516 yon12 17517 yon2 17518 oyon1cl 17523 yonedalem21 17525 yonedalem3a 17526 yonedalem4c 17529 yonedalem22 17530 yonedalem3b 17531 yonedalem3 17532 yonedainv 17533 yonffthlem 17534 |
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