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Theorem fullresc 16280
Description: The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
fullsubc.b 𝐵 = (Base‘𝐶)
fullsubc.h 𝐻 = (Homf𝐶)
fullsubc.c (𝜑𝐶 ∈ Cat)
fullsubc.s (𝜑𝑆𝐵)
fullsubc.d 𝐷 = (𝐶s 𝑆)
fullsubc.e 𝐸 = (𝐶cat (𝐻 ↾ (𝑆 × 𝑆)))
Assertion
Ref Expression
fullresc (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))

Proof of Theorem fullresc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullsubc.h . . . . . 6 𝐻 = (Homf𝐶)
2 fullsubc.b . . . . . 6 𝐵 = (Base‘𝐶)
3 eqid 2609 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
4 fullsubc.s . . . . . . . 8 (𝜑𝑆𝐵)
54adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑆𝐵)
6 simprl 789 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
75, 6sseldd 3568 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝐵)
8 simprr 791 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝑆)
95, 8sseldd 3568 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑦𝐵)
101, 2, 3, 7, 9homfval 16121 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
116, 8ovresd 6677 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
12 fullsubc.e . . . . . . . 8 𝐸 = (𝐶cat (𝐻 ↾ (𝑆 × 𝑆)))
13 fullsubc.c . . . . . . . 8 (𝜑𝐶 ∈ Cat)
141, 2homffn 16122 . . . . . . . . 9 𝐻 Fn (𝐵 × 𝐵)
15 xpss12 5137 . . . . . . . . . 10 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
164, 4, 15syl2anc 690 . . . . . . . . 9 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17 fnssres 5904 . . . . . . . . 9 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1814, 16, 17sylancr 693 . . . . . . . 8 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
1912, 2, 13, 18, 4reschom 16259 . . . . . . 7 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
2019oveqdr 6551 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
2111, 20eqtr3d 2645 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22 fullsubc.d . . . . . . . . . 10 𝐷 = (𝐶s 𝑆)
2322, 2ressbas2 15704 . . . . . . . . 9 (𝑆𝐵𝑆 = (Base‘𝐷))
244, 23syl 17 . . . . . . . 8 (𝜑𝑆 = (Base‘𝐷))
25 fvex 6098 . . . . . . . 8 (Base‘𝐷) ∈ V
2624, 25syl6eqel 2695 . . . . . . 7 (𝜑𝑆 ∈ V)
2722, 3resshom 15847 . . . . . . 7 (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
2826, 27syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
2928oveqdr 6551 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
3010, 21, 293eqtr3rd 2652 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
3130ralrimivva 2953 . . 3 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32 eqid 2609 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
33 eqid 2609 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3412, 2, 13, 18, 4rescbas 16258 . . . 4 (𝜑𝑆 = (Base‘𝐸))
3532, 33, 24, 34homfeq 16123 . . 3 (𝜑 → ((Homf𝐷) = (Homf𝐸) ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
3631, 35mpbird 245 . 2 (𝜑 → (Homf𝐷) = (Homf𝐸))
37 eqid 2609 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
3822, 37ressco 15848 . . . . 5 (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
3926, 38syl 17 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
4012, 2, 13, 18, 4, 37rescco 16261 . . . 4 (𝜑 → (comp‘𝐶) = (comp‘𝐸))
4139, 40eqtr3d 2645 . . 3 (𝜑 → (comp‘𝐷) = (comp‘𝐸))
4241, 36comfeqd 16136 . 2 (𝜑 → (compf𝐷) = (compf𝐸))
4336, 42jca 552 1 (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  wss 3539   × cxp 5026  cres 5030   Fn wfn 5785  cfv 5790  (class class class)co 6527  Basecbs 15641  s cress 15642  Hom chom 15725  compcco 15726  Catccat 16094  Homf chomf 16096  compfccomf 16097  cat cresc 16237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-hom 15739  df-cco 15740  df-homf 16100  df-comf 16101  df-resc 16240
This theorem is referenced by:  resscat  16281  funcres2c  16330  ressffth  16367  funcsetcres2  16512
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