MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsubc Structured version   Visualization version   GIF version

Theorem subsubc 16685
Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
subsubc.d 𝐷 = (𝐶cat 𝐻)
Assertion
Ref Expression
subsubc (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))

Proof of Theorem subsubc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6 (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ∈ (Subcat‘𝐷))
2 eqid 2748 . . . . . 6 (Homf𝐷) = (Homf𝐷)
31, 2subcssc 16672 . . . . 5 (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat (Homf𝐷))
4 subsubc.d . . . . . . 7 𝐷 = (𝐶cat 𝐻)
5 eqid 2748 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
6 subcrcl 16648 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7 id 22 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8 eqidd 2749 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
97, 8subcfn 16673 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
107, 9, 5subcss1 16674 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
114, 5, 6, 9, 10reschomf 16663 . . . . . 6 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf𝐷))
1211breq2d 4804 . . . . 5 (𝐻 ∈ (Subcat‘𝐶) → (𝐽cat 𝐻𝐽cat (Homf𝐷)))
133, 12syl5ibr 236 . . . 4 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat 𝐻))
1413pm4.71rd 670 . . 3 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷))))
15 simpr 479 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat 𝐻)
16 simpl 474 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17 eqid 2748 . . . . . . . . 9 (Homf𝐶) = (Homf𝐶)
1816, 17subcssc 16672 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻cat (Homf𝐶))
19 ssctr 16657 . . . . . . . 8 ((𝐽cat 𝐻𝐻cat (Homf𝐶)) → 𝐽cat (Homf𝐶))
2015, 18, 19syl2anc 696 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐶))
2112biimpa 502 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐷))
2220, 212thd 255 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽cat (Homf𝐶) ↔ 𝐽cat (Homf𝐷)))
2316adantr 472 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
249adantr 472 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
2524adantr 472 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26 eqid 2748 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
27 eqidd 2749 . . . . . . . . . . . 12 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
2815, 27sscfn1 16649 . . . . . . . . . . 11 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
2928, 24, 15ssc1 16653 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
3029sselda 3732 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 16679 . . . . . . . 8 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
3231eleq1d 2812 . . . . . . 7 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
3332ralbidva 3111 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
344oveq1i 6811 . . . . . . . 8 (𝐷cat 𝐽) = ((𝐶cat 𝐻) ↾cat 𝐽)
356adantr 472 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐶 ∈ Cat)
36 dmexg 7250 . . . . . . . . . . 11 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37 dmexg 7250 . . . . . . . . . . 11 (dom 𝐻 ∈ V → dom dom 𝐻 ∈ V)
3836, 37syl 17 . . . . . . . . . 10 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
3938adantr 472 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐻 ∈ V)
4035, 24, 28, 39, 29rescabs 16665 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
4134, 40syl5req 2795 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐶cat 𝐽) = (𝐷cat 𝐽))
4241eleq1d 2812 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐽) ∈ Cat ↔ (𝐷cat 𝐽) ∈ Cat))
4322, 33, 423anbi123d 1536 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat) ↔ (𝐽cat (Homf𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷cat 𝐽) ∈ Cat)))
44 eqid 2748 . . . . . 6 (𝐶cat 𝐽) = (𝐶cat 𝐽)
4517, 26, 44, 35, 28issubc3 16681 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat)))
46 eqid 2748 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
47 eqid 2748 . . . . . 6 (𝐷cat 𝐽) = (𝐷cat 𝐽)
484, 7subccat 16680 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
4948adantr 472 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐷 ∈ Cat)
502, 46, 47, 49, 28issubc3 16681 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat (Homf𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷cat 𝐽) ∈ Cat)))
5143, 45, 503bitr4rd 301 . . . 4 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
5251pm5.32da 676 . . 3 (𝐻 ∈ (Subcat‘𝐶) → ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
5314, 52bitrd 268 . 2 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
54 ancom 465 . 2 ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶)) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻))
5553, 54syl6bb 276 1 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  Vcvv 3328   class class class wbr 4792   × cxp 5252  dom cdm 5254   Fn wfn 6032  cfv 6037  (class class class)co 6801  Basecbs 16030  Catccat 16497  Idccid 16498  Homf chomf 16499  cat cssc 16639  cat cresc 16640  Subcatcsubc 16641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-pm 8014  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-2 11242  df-3 11243  df-4 11244  df-5 11245  df-6 11246  df-7 11247  df-8 11248  df-9 11249  df-n0 11456  df-z 11541  df-dec 11657  df-ndx 16033  df-slot 16034  df-base 16036  df-sets 16037  df-ress 16038  df-hom 16139  df-cco 16140  df-cat 16501  df-cid 16502  df-homf 16503  df-ssc 16642  df-resc 16643  df-subc 16644
This theorem is referenced by:  fldhmsubc  42563  fldhmsubcALTV  42581
  Copyright terms: Public domain W3C validator