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Theorem subsubc 16282
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 2609 . . . . . 6 (Homf𝐷) = (Homf𝐷)
31, 2subcssc 16269 . . . . 5 (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat (Homf𝐷))
4 subsubc.d . . . . . . 7 𝐷 = (𝐶cat 𝐻)
5 eqid 2609 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
6 subcrcl 16245 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7 id 22 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8 eqidd 2610 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
97, 8subcfn 16270 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
107, 9, 5subcss1 16271 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
114, 5, 6, 9, 10reschomf 16260 . . . . . 6 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf𝐷))
1211breq2d 4589 . . . . 5 (𝐻 ∈ (Subcat‘𝐶) → (𝐽cat 𝐻𝐽cat (Homf𝐷)))
133, 12syl5ibr 234 . . . 4 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat 𝐻))
1413pm4.71rd 664 . . 3 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷))))
15 simpr 475 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat 𝐻)
16 simpl 471 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17 eqid 2609 . . . . . . . . 9 (Homf𝐶) = (Homf𝐶)
1816, 17subcssc 16269 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻cat (Homf𝐶))
19 ssctr 16254 . . . . . . . 8 ((𝐽cat 𝐻𝐻cat (Homf𝐶)) → 𝐽cat (Homf𝐶))
2015, 18, 19syl2anc 690 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐶))
2112biimpa 499 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐷))
2220, 212thd 253 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽cat (Homf𝐶) ↔ 𝐽cat (Homf𝐷)))
2316adantr 479 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
249adantr 479 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
2524adantr 479 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26 eqid 2609 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
27 eqidd 2610 . . . . . . . . . . . 12 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
2815, 27sscfn1 16246 . . . . . . . . . . 11 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
2928, 24, 15ssc1 16250 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
3029sselda 3567 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 16276 . . . . . . . 8 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
3231eleq1d 2671 . . . . . . 7 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
3332ralbidva 2967 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
344oveq1i 6537 . . . . . . . 8 (𝐷cat 𝐽) = ((𝐶cat 𝐻) ↾cat 𝐽)
356adantr 479 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐶 ∈ Cat)
36 dmexg 6966 . . . . . . . . . . 11 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
37 dmexg 6966 . . . . . . . . . . 11 (dom 𝐻 ∈ V → dom dom 𝐻 ∈ V)
3836, 37syl 17 . . . . . . . . . 10 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
3938adantr 479 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐻 ∈ V)
4035, 24, 28, 39, 29rescabs 16262 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
4134, 40syl5req 2656 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐶cat 𝐽) = (𝐷cat 𝐽))
4241eleq1d 2671 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐽) ∈ Cat ↔ (𝐷cat 𝐽) ∈ Cat))
4322, 33, 423anbi123d 1390 . . . . 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 2609 . . . . . 6 (𝐶cat 𝐽) = (𝐶cat 𝐽)
4517, 26, 44, 35, 28issubc3 16278 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat)))
46 eqid 2609 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
47 eqid 2609 . . . . . 6 (𝐷cat 𝐽) = (𝐷cat 𝐽)
484, 7subccat 16277 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
4948adantr 479 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐷 ∈ Cat)
502, 46, 47, 49, 28issubc3 16278 . . . . 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 299 . . . 4 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
5251pm5.32da 670 . . 3 (𝐻 ∈ (Subcat‘𝐶) → ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
5314, 52bitrd 266 . 2 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
54 ancom 464 . 2 ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶)) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻))
5553, 54syl6bb 274 1 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172   class class class wbr 4577   × cxp 5026  dom cdm 5028   Fn wfn 5785  cfv 5790  (class class class)co 6527  Basecbs 15641  Catccat 16094  Idccid 16095  Homf chomf 16096  cat cssc 16236  cat cresc 16237  Subcatcsubc 16238
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-fal 1480  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-rmo 2903  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-pm 7724  df-ixp 7772  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-cat 16098  df-cid 16099  df-homf 16100  df-ssc 16239  df-resc 16240  df-subc 16241
This theorem is referenced by:  fldhmsubc  41871  fldhmsubcALTV  41890
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