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Theorem subccatid 16329
 Description: A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subccat.1 𝐷 = (𝐶cat 𝐽)
subccat.j (𝜑𝐽 ∈ (Subcat‘𝐶))
subccatid.1 (𝜑𝐽 Fn (𝑆 × 𝑆))
subccatid.2 1 = (Id‘𝐶)
Assertion
Ref Expression
subccatid (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥𝑆 ↦ ( 1𝑥))))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥   𝑥, 1   𝑥,𝐽   𝑥,𝑆

Proof of Theorem subccatid
Dummy variables 𝑓 𝑔 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subccat.1 . . 3 𝐷 = (𝐶cat 𝐽)
2 eqid 2610 . . 3 (Base‘𝐶) = (Base‘𝐶)
3 subccat.j . . . 4 (𝜑𝐽 ∈ (Subcat‘𝐶))
4 subcrcl 16299 . . . 4 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
53, 4syl 17 . . 3 (𝜑𝐶 ∈ Cat)
6 subccatid.1 . . 3 (𝜑𝐽 Fn (𝑆 × 𝑆))
73, 6, 2subcss1 16325 . . 3 (𝜑𝑆 ⊆ (Base‘𝐶))
81, 2, 5, 6, 7rescbas 16312 . 2 (𝜑𝑆 = (Base‘𝐷))
91, 2, 5, 6, 7reschom 16313 . 2 (𝜑𝐽 = (Hom ‘𝐷))
10 eqid 2610 . . 3 (comp‘𝐶) = (comp‘𝐶)
111, 2, 5, 6, 7, 10rescco 16315 . 2 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
12 ovex 6577 . . . 4 (𝐶cat 𝐽) ∈ V
131, 12eqeltri 2684 . . 3 𝐷 ∈ V
1413a1i 11 . 2 (𝜑𝐷 ∈ V)
15 biid 250 . 2 (((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧))) ↔ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧))))
163adantr 480 . . 3 ((𝜑𝑥𝑆) → 𝐽 ∈ (Subcat‘𝐶))
176adantr 480 . . 3 ((𝜑𝑥𝑆) → 𝐽 Fn (𝑆 × 𝑆))
18 simpr 476 . . 3 ((𝜑𝑥𝑆) → 𝑥𝑆)
19 subccatid.2 . . 3 1 = (Id‘𝐶)
2016, 17, 18, 19subcidcl 16327 . 2 ((𝜑𝑥𝑆) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
21 eqid 2610 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
225adantr 480 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝐶 ∈ Cat)
237adantr 480 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑆 ⊆ (Base‘𝐶))
24 simpr1l 1111 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑤𝑆)
2523, 24sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑤 ∈ (Base‘𝐶))
26 simpr1r 1112 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑥𝑆)
2723, 26sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑥 ∈ (Base‘𝐶))
283adantr 480 . . . . 5 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝐽 ∈ (Subcat‘𝐶))
296adantr 480 . . . . 5 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝐽 Fn (𝑆 × 𝑆))
3028, 29, 21, 24, 26subcss2 16326 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (𝑤𝐽𝑥) ⊆ (𝑤(Hom ‘𝐶)𝑥))
31 simpr31 1144 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑤𝐽𝑥))
3230, 31sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑓 ∈ (𝑤(Hom ‘𝐶)𝑥))
332, 21, 19, 22, 25, 10, 27, 32catlid 16167 . 2 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (( 1𝑥)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
34 simpr2l 1113 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑦𝑆)
3523, 34sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑦 ∈ (Base‘𝐶))
3628, 29, 21, 26, 34subcss2 16326 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦))
37 simpr32 1145 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑥𝐽𝑦))
3836, 37sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦))
392, 21, 19, 22, 27, 10, 35, 38catrid 16168 . 2 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)( 1𝑥)) = 𝑔)
4028, 29, 24, 10, 26, 34, 31, 37subccocl 16328 . 2 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓) ∈ (𝑤𝐽𝑦))
41 simpr2r 1114 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑧𝑆)
4223, 41sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → 𝑧 ∈ (Base‘𝐶))
4328, 29, 21, 34, 41subcss2 16326 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (𝑦𝐽𝑧) ⊆ (𝑦(Hom ‘𝐶)𝑧))
44 simpr33 1146 . . . 4 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → ∈ (𝑦𝐽𝑧))
4543, 44sseldd 3569 . . 3 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → ∈ (𝑦(Hom ‘𝐶)𝑧))
462, 21, 10, 22, 25, 27, 35, 32, 38, 42, 45catass 16170 . 2 ((𝜑 ∧ ((𝑤𝑆𝑥𝑆) ∧ (𝑦𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑤𝐽𝑥) ∧ 𝑔 ∈ (𝑥𝐽𝑦) ∧ ∈ (𝑦𝐽𝑧)))) → (((⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔)(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑧)𝑓) = ((⟨𝑤, 𝑦⟩(comp‘𝐶)𝑧)(𝑔(⟨𝑤, 𝑥⟩(comp‘𝐶)𝑦)𝑓)))
478, 9, 11, 14, 15, 20, 33, 39, 40, 46iscatd2 16165 1 (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥𝑆 ↦ ( 1𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   ↦ cmpt 4643   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149   ↾cat cresc 16291  Subcatcsubc 16292 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-homf 16154  df-ssc 16293  df-resc 16294  df-subc 16295 This theorem is referenced by:  subcid  16330  subccat  16331
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