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Mirrors > Home > MPE Home > Th. List > rescco | Structured version Visualization version GIF version |
Description: Composition in 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) |
rescco.o | ⊢ · = (comp‘𝐶) |
Ref | Expression |
---|---|
rescco | ⊢ (𝜑 → · = (comp‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccoid 16684 | . . 3 ⊢ comp = Slot (comp‘ndx) | |
2 | 1nn0 11907 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
3 | 4nn 11714 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
4 | 2, 3 | decnncl 12112 | . . . . . 6 ⊢ ;14 ∈ ℕ |
5 | 4 | nnrei 11641 | . . . . 5 ⊢ ;14 ∈ ℝ |
6 | 4nn0 11910 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
7 | 5nn 11717 | . . . . . 6 ⊢ 5 ∈ ℕ | |
8 | 4lt5 11808 | . . . . . 6 ⊢ 4 < 5 | |
9 | 2, 6, 7, 8 | declt 12120 | . . . . 5 ⊢ ;14 < ;15 |
10 | 5, 9 | gtneii 10746 | . . . 4 ⊢ ;15 ≠ ;14 |
11 | ccondx 16683 | . . . . 5 ⊢ (comp‘ndx) = ;15 | |
12 | homndx 16681 | . . . . 5 ⊢ (Hom ‘ndx) = ;14 | |
13 | 11, 12 | neeq12i 3082 | . . . 4 ⊢ ((comp‘ndx) ≠ (Hom ‘ndx) ↔ ;15 ≠ ;14) |
14 | 10, 13 | mpbir 233 | . . 3 ⊢ (comp‘ndx) ≠ (Hom ‘ndx) |
15 | 1, 14 | setsnid 16533 | . 2 ⊢ (comp‘(𝐶 ↾s 𝑆)) = (comp‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
16 | rescbas.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
17 | rescbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
18 | 17 | fvexi 6678 | . . . . 5 ⊢ 𝐵 ∈ V |
19 | 18 | ssex 5217 | . . . 4 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ V) |
20 | 16, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
21 | eqid 2821 | . . . 4 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆) | |
22 | rescco.o | . . . 4 ⊢ · = (comp‘𝐶) | |
23 | 21, 22 | ressco 16686 | . . 3 ⊢ (𝑆 ∈ V → · = (comp‘(𝐶 ↾s 𝑆))) |
24 | 20, 23 | syl 17 | . 2 ⊢ (𝜑 → · = (comp‘(𝐶 ↾s 𝑆))) |
25 | rescbas.d | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
26 | rescbas.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
27 | rescbas.h | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
28 | 25, 26, 20, 27 | rescval2 17092 | . . 3 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
29 | 28 | fveq2d 6668 | . 2 ⊢ (𝜑 → (comp‘𝐷) = (comp‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
30 | 15, 24, 29 | 3eqtr4a 2882 | 1 ⊢ (𝜑 → · = (comp‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ⊆ wss 3935 〈cop 4566 × cxp 5547 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 1c1 10532 4c4 11688 5c5 11689 ;cdc 12092 ndxcnx 16474 sSet csts 16475 Basecbs 16477 ↾s cress 16478 Hom chom 16570 compcco 16571 ↾cat cresc 17072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-hom 16583 df-cco 16584 df-resc 17075 |
This theorem is referenced by: subccatid 17110 issubc3 17113 fullresc 17115 funcres 17160 funcres2b 17161 rngccofval 44235 ringccofval 44281 |
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