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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.) (Proof shortened by AV, 13-Oct-2024.) |
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 17366 | . . 3 ⊢ comp = Slot (comp‘ndx) | |
2 | slotsbhcdif 17367 | . . . 4 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) | |
3 | simp3 1137 | . . . . 5 ⊢ (((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) → (Hom ‘ndx) ≠ (comp‘ndx)) | |
4 | 3 | necomd 2995 | . . . 4 ⊢ (((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) → (comp‘ndx) ≠ (Hom ‘ndx)) |
5 | 2, 4 | ax-mp 5 | . . 3 ⊢ (comp‘ndx) ≠ (Hom ‘ndx) |
6 | 1, 5 | setsnid 17149 | . 2 ⊢ (comp‘(𝐶 ↾s 𝑆)) = (comp‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
7 | rescbas.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
8 | rescbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
9 | 8 | fvexi 6905 | . . . . 5 ⊢ 𝐵 ∈ V |
10 | 9 | ssex 5321 | . . . 4 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ V) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
12 | eqid 2731 | . . . 4 ⊢ (𝐶 ↾s 𝑆) = (𝐶 ↾s 𝑆) | |
13 | rescco.o | . . . 4 ⊢ · = (comp‘𝐶) | |
14 | 12, 13 | ressco 17372 | . . 3 ⊢ (𝑆 ∈ V → · = (comp‘(𝐶 ↾s 𝑆))) |
15 | 11, 14 | syl 17 | . 2 ⊢ (𝜑 → · = (comp‘(𝐶 ↾s 𝑆))) |
16 | rescbas.d | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
17 | rescbas.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
18 | rescbas.h | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
19 | 16, 17, 11, 18 | rescval2 17782 | . . 3 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
20 | 19 | fveq2d 6895 | . 2 ⊢ (𝜑 → (comp‘𝐷) = (comp‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
21 | 6, 15, 20 | 3eqtr4a 2797 | 1 ⊢ (𝜑 → · = (comp‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ⊆ wss 3948 〈cop 4634 × cxp 5674 Fn wfn 6538 ‘cfv 6543 (class class class)co 7412 sSet csts 17103 ndxcnx 17133 Basecbs 17151 ↾s cress 17180 Hom chom 17215 compcco 17216 ↾cat cresc 17762 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-hom 17228 df-cco 17229 df-resc 17765 |
This theorem is referenced by: subccatid 17803 issubc3 17806 fullresc 17808 funcres 17853 funcres2b 17854 rngccofval 20518 ringccofval 20547 |
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