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Theorem subsubc 16824
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 2797 . . . . . 6 (Homf𝐷) = (Homf𝐷)
31, 2subcssc 16811 . . . . 5 (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat (Homf𝐷))
4 subsubc.d . . . . . . 7 𝐷 = (𝐶cat 𝐻)
5 eqid 2797 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
6 subcrcl 16787 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7 id 22 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8 eqidd 2798 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
97, 8subcfn 16812 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
107, 9, 5subcss1 16813 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
114, 5, 6, 9, 10reschomf 16802 . . . . . 6 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf𝐷))
1211breq2d 4853 . . . . 5 (𝐻 ∈ (Subcat‘𝐶) → (𝐽cat 𝐻𝐽cat (Homf𝐷)))
133, 12syl5ibr 238 . . . 4 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat 𝐻))
1413pm4.71rd 559 . . 3 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷))))
15 simpr 478 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat 𝐻)
16 simpl 475 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17 eqid 2797 . . . . . . . . 9 (Homf𝐶) = (Homf𝐶)
1816, 17subcssc 16811 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻cat (Homf𝐶))
19 ssctr 16796 . . . . . . . 8 ((𝐽cat 𝐻𝐻cat (Homf𝐶)) → 𝐽cat (Homf𝐶))
2015, 18, 19syl2anc 580 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐶))
2112biimpa 469 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐷))
2220, 212thd 257 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽cat (Homf𝐶) ↔ 𝐽cat (Homf𝐷)))
2316adantr 473 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
249adantr 473 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
2524adantr 473 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26 eqid 2797 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
27 eqidd 2798 . . . . . . . . . . . 12 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
2815, 27sscfn1 16788 . . . . . . . . . . 11 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
2928, 24, 15ssc1 16792 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
3029sselda 3796 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 16818 . . . . . . . 8 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
3231eleq1d 2861 . . . . . . 7 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
3332ralbidva 3164 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
344oveq1i 6886 . . . . . . . 8 (𝐷cat 𝐽) = ((𝐶cat 𝐻) ↾cat 𝐽)
356adantr 473 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐶 ∈ Cat)
36 dmexg 7329 . . . . . . . . . . 11 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
3736dmexd 7331 . . . . . . . . . 10 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
3837adantr 473 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐻 ∈ V)
3935, 24, 28, 38, 29rescabs 16804 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
4034, 39syl5req 2844 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐶cat 𝐽) = (𝐷cat 𝐽))
4140eleq1d 2861 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐽) ∈ Cat ↔ (𝐷cat 𝐽) ∈ Cat))
4222, 33, 413anbi123d 1561 . . . . 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)))
43 eqid 2797 . . . . . 6 (𝐶cat 𝐽) = (𝐶cat 𝐽)
4417, 26, 43, 35, 28issubc3 16820 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat)))
45 eqid 2797 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
46 eqid 2797 . . . . . 6 (𝐷cat 𝐽) = (𝐷cat 𝐽)
474, 7subccat 16819 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
4847adantr 473 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐷 ∈ Cat)
492, 45, 46, 48, 28issubc3 16820 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat (Homf𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷cat 𝐽) ∈ Cat)))
5042, 44, 493bitr4rd 304 . . . 4 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
5150pm5.32da 575 . . 3 (𝐻 ∈ (Subcat‘𝐶) → ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
5214, 51bitrd 271 . 2 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
53 ancom 453 . 2 ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶)) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻))
5452, 53syl6bb 279 1 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  Vcvv 3383   class class class wbr 4841   × cxp 5308  dom cdm 5310   Fn wfn 6094  cfv 6099  (class class class)co 6876  Basecbs 16181  Catccat 16636  Idccid 16637  Homf chomf 16638  cat cssc 16778  cat cresc 16779  Subcatcsubc 16780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-er 7980  df-pm 8096  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-3 11373  df-4 11374  df-5 11375  df-6 11376  df-7 11377  df-8 11378  df-9 11379  df-n0 11577  df-z 11663  df-dec 11780  df-ndx 16184  df-slot 16185  df-base 16187  df-sets 16188  df-ress 16189  df-hom 16288  df-cco 16289  df-cat 16640  df-cid 16641  df-homf 16642  df-ssc 16781  df-resc 16782  df-subc 16783
This theorem is referenced by:  fldhmsubc  42871  fldhmsubcALTV  42889
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