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Theorem subsubc 16560
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 2651 . . . . . 6 (Homf𝐷) = (Homf𝐷)
31, 2subcssc 16547 . . . . 5 (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat (Homf𝐷))
4 subsubc.d . . . . . . 7 𝐷 = (𝐶cat 𝐻)
5 eqid 2651 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
6 subcrcl 16523 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7 id 22 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8 eqidd 2652 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
97, 8subcfn 16548 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
107, 9, 5subcss1 16549 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
114, 5, 6, 9, 10reschomf 16538 . . . . . 6 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf𝐷))
1211breq2d 4697 . . . . 5 (𝐻 ∈ (Subcat‘𝐶) → (𝐽cat 𝐻𝐽cat (Homf𝐷)))
133, 12syl5ibr 236 . . . 4 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat 𝐻))
1413pm4.71rd 668 . . 3 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷))))
15 simpr 476 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat 𝐻)
16 simpl 472 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17 eqid 2651 . . . . . . . . 9 (Homf𝐶) = (Homf𝐶)
1816, 17subcssc 16547 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻cat (Homf𝐶))
19 ssctr 16532 . . . . . . . 8 ((𝐽cat 𝐻𝐻cat (Homf𝐶)) → 𝐽cat (Homf𝐶))
2015, 18, 19syl2anc 694 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐶))
2112biimpa 500 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐷))
2220, 212thd 255 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽cat (Homf𝐶) ↔ 𝐽cat (Homf𝐷)))
2316adantr 480 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
249adantr 480 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
2524adantr 480 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26 eqid 2651 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
27 eqidd 2652 . . . . . . . . . . . 12 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
2815, 27sscfn1 16524 . . . . . . . . . . 11 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
2928, 24, 15ssc1 16528 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
3029sselda 3636 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 16554 . . . . . . . 8 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
3231eleq1d 2715 . . . . . . 7 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
3332ralbidva 3014 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
344oveq1i 6700 . . . . . . . 8 (𝐷cat 𝐽) = ((𝐶cat 𝐻) ↾cat 𝐽)
356adantr 480 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐶 ∈ Cat)
36 dmexg 7139 . . . . . . . . . . 11 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37 dmexg 7139 . . . . . . . . . . 11 (dom 𝐻 ∈ V → dom dom 𝐻 ∈ V)
3836, 37syl 17 . . . . . . . . . 10 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
3938adantr 480 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐻 ∈ V)
4035, 24, 28, 39, 29rescabs 16540 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
4134, 40syl5req 2698 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐶cat 𝐽) = (𝐷cat 𝐽))
4241eleq1d 2715 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐽) ∈ Cat ↔ (𝐷cat 𝐽) ∈ Cat))
4322, 33, 423anbi123d 1439 . . . . 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 2651 . . . . . 6 (𝐶cat 𝐽) = (𝐶cat 𝐽)
4517, 26, 44, 35, 28issubc3 16556 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat)))
46 eqid 2651 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
47 eqid 2651 . . . . . 6 (𝐷cat 𝐽) = (𝐷cat 𝐽)
484, 7subccat 16555 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
4948adantr 480 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐷 ∈ Cat)
502, 46, 47, 49, 28issubc3 16556 . . . . 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 674 . . 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 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231   class class class wbr 4685   × cxp 5141  dom cdm 5143   Fn wfn 5921  cfv 5926  (class class class)co 6690  Basecbs 15904  Catccat 16372  Idccid 16373  Homf chomf 16374  cat cssc 16514  cat cresc 16515  Subcatcsubc 16516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-homf 16378  df-ssc 16517  df-resc 16518  df-subc 16519
This theorem is referenced by:  fldhmsubc  42409  fldhmsubcALTV  42427
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