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Theorem subsubc 17799
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 17786 . . . . 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 17759 . . . . . . 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 17787 . . . . . . 7 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐻 Fn (dom dom 𝐻 Γ— dom dom 𝐻))
107, 9, 5subcss1 17788 . . . . . . 7 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ dom dom 𝐻 βŠ† (Baseβ€˜πΆ))
114, 5, 6, 9, 10reschomf 17775 . . . . . 6 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ 𝐻 = (Homf β€˜π·))
1211breq2d 5159 . . . . 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 17786 . . . . . . . 8 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐻 βŠ†cat (Homf β€˜πΆ))
19 ssctr 17768 . . . . . . . 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 17760 . . . . . . . . . . 11 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ 𝐽 Fn (dom dom 𝐽 Γ— dom dom 𝐽))
2928, 24, 15ssc1 17764 . . . . . . . . . 10 ((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) β†’ dom dom 𝐽 βŠ† dom dom 𝐻)
3029sselda 3981 . . . . . . . . 9 (((𝐻 ∈ (Subcatβ€˜πΆ) ∧ 𝐽 βŠ†cat 𝐻) ∧ π‘₯ ∈ dom dom 𝐽) β†’ π‘₯ ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 17793 . . . . . . . 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 7415 . . . . . . . 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 7890 . . . . . . . . . . 11 (𝐻 ∈ (Subcatβ€˜πΆ) β†’ dom 𝐻 ∈ V)
3736dmexd 7892 . . . . . . . . . 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 17778 . . . . . . . 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 17795 . . . . 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 17794 . . . . . . 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 17795 . . . . 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 5147   Γ— cxp 5673  dom cdm 5675   Fn wfn 6535  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Catccat 17604  Idccid 17605  Homf chomf 17606   βŠ†cat cssc 17750   β†Ύcat cresc 17751  Subcatcsubc 17752
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-hom 17217  df-cco 17218  df-cat 17608  df-cid 17609  df-homf 17610  df-ssc 17753  df-resc 17754  df-subc 17755
This theorem is referenced by:  fldhmsubc  46935  fldhmsubcALTV  46953
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