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Mirrors > Home > MPE Home > Th. List > rescbas | Structured version Visualization version GIF version |
Description: Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rescbas.d | ⊢ 𝐷 = (𝐶 ↾cat 𝐻) |
rescbas.b | ⊢ 𝐵 = (Base‘𝐶) |
rescbas.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
rescbas.h | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
rescbas.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Ref | Expression |
---|---|
rescbas | ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baseid 15919 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
2 | 1re 10039 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1nn 11031 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | 4nn0 11311 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
5 | 1nn0 11308 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 11681 | . . . . . 6 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 11546 | . . . . 5 ⊢ 1 < ;14 |
8 | 2, 7 | ltneii 10150 | . . . 4 ⊢ 1 ≠ ;14 |
9 | basendx 15923 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
10 | homndx 16074 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
11 | 9, 10 | neeq12i 2860 | . . . 4 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14) |
12 | 8, 11 | mpbir 221 | . . 3 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
13 | 1, 12 | setsnid 15915 | . 2 ⊢ (Base‘(𝐶 ↾s 𝑆)) = (Base‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
14 | rescbas.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
15 | eqid 2622 | . . . 4 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆) | |
16 | rescbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
17 | 15, 16 | ressbas2 15931 | . . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘(𝐶 ↾s 𝑆))) |
18 | 14, 17 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 = (Base‘(𝐶 ↾s 𝑆))) |
19 | rescbas.d | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
20 | rescbas.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
21 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝐶) ∈ V | |
22 | 16, 21 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
23 | 22 | ssex 4802 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ V) |
24 | 14, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
25 | rescbas.h | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
26 | 19, 20, 24, 25 | rescval2 16488 | . . 3 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
27 | 26 | fveq2d 6195 | . 2 ⊢ (𝜑 → (Base‘𝐷) = (Base‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
28 | 13, 18, 27 | 3eqtr4a 2682 | 1 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 〈cop 4183 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 1c1 9937 4c4 11072 ;cdc 11493 ndxcnx 15854 sSet csts 15855 Basecbs 15857 ↾s cress 15858 Hom chom 15952 ↾cat cresc 16468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-hom 15966 df-resc 16471 |
This theorem is referenced by: reschomf 16491 subccatid 16506 issubc3 16509 fullresc 16511 funcres 16556 funcres2b 16557 funcres2 16558 rngcbas 41965 ringcbas 42011 |
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