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Theorem subsubc 17807
Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
subsubc.d 𝐷 = (𝐢 β†Ύcat 𝐻)
Assertion
Ref Expression
subsubc (𝐻 ∈ (Subcatβ€˜πΆ) β†’ (𝐽 ∈ (Subcatβ€˜π·) ↔ (𝐽 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻)))
Proof of Theorem subsubc
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6 (𝐽 ∈ (Subcatβ€˜π·) β†’ 𝐽 ∈ (Subcatβ€˜π·))
2 eqid 2732 . . . . . 6 (Homf β€˜π·) = (Homf β€˜π·)
31, 2subcssc 17794 . . . . 5 (𝐽 ∈ (Subcatβ€˜π·) β†’ 𝐽 βŠ†cat (Homf β€˜π·))
4 subsubc.d . . . . . . 7 𝐷 = (𝐢 β†Ύcat 𝐻)
5 eqid 2732 . . . . . . 7 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
6 subcrcl 17767 . . . . . . 7 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐢 ∈ Cat)
7 id 22 . . . . . . . 8 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐻 ∈ (Subcatβ€˜πΆ))
8 eqidd 2733 . . . . . . . 8 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ dom dom 𝐻 = dom dom 𝐻)
97, 8subcfn 17795 . . . . . . 7 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐻 Fn (dom dom 𝐻 Γ— dom dom 𝐻))
107, 9, 5subcss1 17796 . . . . . . 7 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ dom dom 𝐻 βŠ† (Baseβ€˜πΆ))
114, 5, 6, 9, 10reschomf 17783 . . . . . 6 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐻 = (Homf β€˜π·))
1211breq2d 5160 . . . . 5 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ (𝐽 βŠ†cat 𝐻 ↔ 𝐽 βŠ†cat (Homf β€˜π·)))
133, 12imbitrrid 245 . . . 4 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ (𝐽 ∈ (Subcatβ€˜π·) β†’ 𝐽 βŠ†cat 𝐻))
1413pm4.71rd 563 . . 3 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ (𝐽 ∈ (Subcatβ€˜π·) ↔ (𝐽 βŠ†cat 𝐻 ∧ 𝐽 ∈ (Subcatβ€˜π·))))
15 simpr 485 . . . . . . . 8 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐽 βŠ†cat 𝐻)
16 simpl 483 . . . . . . . . 9 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐻 ∈ (Subcatβ€˜πΆ))
17 eqid 2732 . . . . . . . . 9 (Homf β€˜πΆ) = (Homf β€˜πΆ)
1816, 17subcssc 17794 . . . . . . . 8 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐻 βŠ†cat (Homf β€˜πΆ))
19 ssctr 17776 . . . . . . . 8 ((𝐽 βŠ†cat 𝐻 ∧ 𝐻 βŠ†cat (Homf β€˜πΆ)) β†’ 𝐽 βŠ†cat (Homf β€˜πΆ))
2015, 18, 19syl2anc 584 . . . . . . 7 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐽 βŠ†cat (Homf β€˜πΆ))
2112biimpa 477 . . . . . . 7 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐽 βŠ†cat (Homf β€˜π·))
2220, 212thd 264 . . . . . 6 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ (𝐽 βŠ†cat (Homf β€˜πΆ) ↔ 𝐽 βŠ†cat (Homf β€˜π·)))
2316adantr 481 . . . . . . . . 9 (((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) ∧ π‘₯ ∈ dom dom 𝐽) β†’ 𝐻 ∈ (Subcatβ€˜πΆ))
249adantr 481 . . . . . . . . . 10 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐻 Fn (dom dom 𝐻 Γ— dom dom 𝐻))
2524adantr 481 . . . . . . . . 9 (((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) ∧ π‘₯ ∈ dom dom 𝐽) β†’ 𝐻 Fn (dom dom 𝐻 Γ— dom dom 𝐻))
26 eqid 2732 . . . . . . . . 9 (Idβ€˜πΆ) = (Idβ€˜πΆ)
27 eqidd 2733 . . . . . . . . . . . 12 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ dom dom 𝐽 = dom dom 𝐽)
2815, 27sscfn1 17768 . . . . . . . . . . 11 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐽 Fn (dom dom 𝐽 Γ— dom dom 𝐽))
2928, 24, 15ssc1 17772 . . . . . . . . . 10 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ dom dom 𝐽 βŠ† dom dom 𝐻)
3029sselda 3982 . . . . . . . . 9 (((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) ∧ π‘₯ ∈ dom dom 𝐽) β†’ π‘₯ ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 17801 . . . . . . . 8 (((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) ∧ π‘₯ ∈ dom dom 𝐽) β†’ ((Idβ€˜πΆ)β€˜π‘₯) = ((Idβ€˜π·)β€˜π‘₯))
3231eleq1d 2818 . . . . . . 7 (((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) ∧ π‘₯ ∈ dom dom 𝐽) β†’ (((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ↔ ((Idβ€˜π·)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯)))
3332ralbidva 3175 . . . . . 6 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ (βˆ€π‘₯ ∈ dom dom 𝐽((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ↔ βˆ€π‘₯ ∈ dom dom 𝐽((Idβ€˜π·)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯)))
344oveq1i 7421 . . . . . . . 8 (𝐷 β†Ύcat 𝐽) = ((𝐢 β†Ύcat 𝐻) β†Ύcat 𝐽)
356adantr 481 . . . . . . . . 9 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐢 ∈ Cat)
36 dmexg 7896 . . . . . . . . . . 11 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ dom 𝐻 ∈ V)
3736dmexd 7898 . . . . . . . . . 10 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ dom dom 𝐻 ∈ V)
3837adantr 481 . . . . . . . . 9 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ dom dom 𝐻 ∈ V)
3935, 24, 28, 38, 29rescabs 17786 . . . . . . . 8 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ ((𝐢 β†Ύcat 𝐻) β†Ύcat 𝐽) = (𝐢 β†Ύcat 𝐽))
4034, 39eqtr2id 2785 . . . . . . 7 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ (𝐢 β†Ύcat 𝐽) = (𝐷 β†Ύcat 𝐽))
4140eleq1d 2818 . . . . . 6 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ ((𝐢 β†Ύcat 𝐽) ∈ Cat ↔ (𝐷 β†Ύcat 𝐽) ∈ Cat))
4222, 33, 413anbi123d 1436 . . . . 5 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ ((𝐽 βŠ†cat (Homf β€˜πΆ) ∧ βˆ€π‘₯ ∈ dom dom 𝐽((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ (𝐢 β†Ύcat 𝐽) ∈ Cat) ↔ (𝐽 βŠ†cat (Homf β€˜π·) ∧ βˆ€π‘₯ ∈ dom dom 𝐽((Idβ€˜π·)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ (𝐷 β†Ύcat 𝐽) ∈ Cat)))
43 eqid 2732 . . . . . 6 (𝐢 β†Ύcat 𝐽) = (𝐢 β†Ύcat 𝐽)
4417, 26, 43, 35, 28issubc3 17803 . . . . 5 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ (𝐽 ∈ (Subcatβ€˜πΆ) ↔ (𝐽 βŠ†cat (Homf β€˜πΆ) ∧ βˆ€π‘₯ ∈ dom dom 𝐽((Idβ€˜πΆ)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ (𝐢 β†Ύcat 𝐽) ∈ Cat)))
45 eqid 2732 . . . . . 6 (Idβ€˜π·) = (Idβ€˜π·)
46 eqid 2732 . . . . . 6 (𝐷 β†Ύcat 𝐽) = (𝐷 β†Ύcat 𝐽)
474, 7subccat 17802 . . . . . . 7 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐷 ∈ Cat)
4847adantr 481 . . . . . 6 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐷 ∈ Cat)
492, 45, 46, 48, 28issubc3 17803 . . . . 5 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ (𝐽 ∈ (Subcatβ€˜π·) ↔ (𝐽 βŠ†cat (Homf β€˜π·) ∧ βˆ€π‘₯ ∈ dom dom 𝐽((Idβ€˜π·)β€˜π‘₯) ∈ (π‘₯𝐽π‘₯) ∧ (𝐷 β†Ύcat 𝐽) ∈ Cat)))
5042, 44, 493bitr4rd 311 . . . 4 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ (𝐽 ∈ (Subcatβ€˜π·) ↔ 𝐽 ∈ (Subcatβ€˜πΆ)))
5150pm5.32da 579 . . 3 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ ((𝐽 βŠ†cat 𝐻 ∧ 𝐽 ∈ (Subcatβ€˜π·)) ↔ (𝐽 βŠ†cat 𝐻 ∧ 𝐽 ∈ (Subcatβ€˜πΆ))))
5214, 51bitrd 278 . 2 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ (𝐽 ∈ (Subcatβ€˜π·) ↔ (𝐽 βŠ†cat 𝐻 ∧ 𝐽 ∈ (Subcatβ€˜πΆ))))
5352biancomd 464 1 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ (𝐽 ∈ (Subcatβ€˜π·) ↔ (𝐽 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   class class class wbr 5148   Γ— cxp 5674  dom cdm 5676   Fn wfn 6538  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  Catccat 17612  Idccid 17613  Homf chomf 17614   βŠ†cat cssc 17758   β†Ύcat cresc 17759  Subcatcsubc 17760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-hom 17225  df-cco 17226  df-cat 17616  df-cid 17617  df-homf 17618  df-ssc 17761  df-resc 17762  df-subc 17763
This theorem is referenced by:  fldhmsubc  47071  fldhmsubcALTV  47089
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