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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 | baseid 16535 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
3 | 1re 10630 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
4 | 1nn 11636 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
5 | 4nn0 11904 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
6 | 1nn0 11901 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
7 | 1lt10 12225 | . . . . . . . . 9 ⊢ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 12124 | . . . . . . . 8 ⊢ 1 < ;14 |
9 | 3, 8 | ltneii 10742 | . . . . . . 7 ⊢ 1 ≠ ;14 |
10 | basendx 16539 | . . . . . . . 8 ⊢ (Base‘ndx) = 1 | |
11 | homndx 16679 | . . . . . . . 8 ⊢ (Hom ‘ndx) = ;14 | |
12 | 10, 11 | neeq12i 3053 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
13 | 9, 12 | mpbir 234 | . . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
14 | 2, 13 | setsnid 16531 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝐶)〉)) |
15 | 5nn 11711 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ | |
16 | 4lt5 11802 | . . . . . . . . . 10 ⊢ 4 < 5 | |
17 | 6, 5, 15, 16 | declt 12114 | . . . . . . . . 9 ⊢ ;14 < ;15 |
18 | 4nn 11708 | . . . . . . . . . . . 12 ⊢ 4 ∈ ℕ | |
19 | 6, 18 | decnncl 12106 | . . . . . . . . . . 11 ⊢ ;14 ∈ ℕ |
20 | 19 | nnrei 11634 | . . . . . . . . . 10 ⊢ ;14 ∈ ℝ |
21 | 6, 15 | decnncl 12106 | . . . . . . . . . . 11 ⊢ ;15 ∈ ℕ |
22 | 21 | nnrei 11634 | . . . . . . . . . 10 ⊢ ;15 ∈ ℝ |
23 | 3, 20, 22 | lttri 10755 | . . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15) |
24 | 8, 17, 23 | mp2an 691 | . . . . . . . 8 ⊢ 1 < ;15 |
25 | 3, 24 | ltneii 10742 | . . . . . . 7 ⊢ 1 ≠ ;15 |
26 | ccondx 16681 | . . . . . . . 8 ⊢ (comp‘ndx) = ;15 | |
27 | 10, 26 | neeq12i 3053 | . . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15) |
28 | 25, 27 | mpbir 234 | . . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx) |
29 | 2, 28 | setsnid 16531 | . . . . 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)))〉)) |
30 | 14, 29 | eqtri 2821 | . . . 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)))〉)) |
31 | eqid 2798 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
32 | eqid 2798 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
33 | eqid 2798 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
34 | oppcbas.1 | . . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶) | |
35 | 31, 32, 33, 34 | oppcval 16975 | . . . . 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)))〉)) |
36 | 35 | fveq2d 6649 | . . . 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)))〉))) |
37 | 30, 36 | eqtr4id 2852 | . . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
38 | base0 16528 | . . . 4 ⊢ ∅ = (Base‘∅) | |
39 | fvprc 6638 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅) | |
40 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅) | |
41 | 34, 40 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅) |
42 | 41 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅)) |
43 | 38, 39, 42 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂)) |
44 | 37, 43 | pm2.61i 185 | . 2 ⊢ (Base‘𝐶) = (Base‘𝑂) |
45 | 1, 44 | eqtri 2821 | 1 ⊢ 𝐵 = (Base‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 〈cop 4531 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 tpos ctpos 7874 1c1 10527 < clt 10664 4c4 11682 5c5 11683 ;cdc 12086 ndxcnx 16472 sSet csts 16473 Basecbs 16475 Hom chom 16568 compcco 16569 oppCatcoppc 16973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-hom 16581 df-cco 16582 df-oppc 16974 |
This theorem is referenced by: oppccatid 16981 oppchomf 16982 2oppcbas 16985 2oppccomf 16987 oppccomfpropd 16989 isepi 17002 epii 17005 oppcsect 17040 oppcsect2 17041 oppcinv 17042 oppciso 17043 sectepi 17046 episect 17047 funcoppc 17137 fulloppc 17184 fthoppc 17185 fthepi 17190 hofcl 17501 yon11 17506 yon12 17507 yon2 17508 oyon1cl 17513 yonedalem21 17515 yonedalem3a 17516 yonedalem4c 17519 yonedalem22 17520 yonedalem3b 17521 yonedalem3 17522 yonedainv 17523 yonffthlem 17524 |
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