Theorem subsubc 15259

Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypothesis

Ref	Expression
subsubc.d	|- D = ( C |`cat H )

Assertion

Ref	Expression
subsubc	|- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) ) )

Proof of Theorem subsubc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 |- ( J e. (Subcat ` D ) -> J e. (Subcat ` D ) )
2		eqid 2382	. . . . . 6 |- ( Hom f ` D ) = ( Hom f ` D )
3	1, 2	subcssc 15246	. . . . 5 |- ( J e. (Subcat ` D ) -> J C_cat ( Hom f ` D ) )
4		subsubc.d	. . . . . . 7 |- D = ( C |`cat H )
5		eqid 2382	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
6		subcrcl 15222	. . . . . . 7 |- ( H e. (Subcat ` C ) -> C e. Cat )
7		id 22	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> H e. (Subcat ` C ) )
8		eqidd 2383	. . . . . . . 8 |- ( H e. (Subcat ` C ) -> dom dom H = dom dom H )
9	7, 8	subcfn 15247	. . . . . . 7 |- ( H e. (Subcat ` C ) -> H Fn ( dom dom H X. dom dom H ) )
10	7, 9, 5	subcss1 15248	. . . . . . 7 |- ( H e. (Subcat ` C ) -> dom dom H C_ ( Base ` C ) )
11	4, 5, 6, 9, 10	reschomf 15237	. . . . . 6 |- ( H e. (Subcat ` C ) -> H = ( Hom f ` D ) )
12	11	breq2d 4379	. . . . 5 |- ( H e. (Subcat ` C ) -> ( J C_cat H <-> J C_cat ( Hom f ` D ) ) )
13	3, 12	syl5ibr 221	. . . 4 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) -> J C_cat H ) )
14	13	pm4.71rd 633	. . 3 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` D ) ) ) )
15		simpr 459	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat H )
16		simpl 455	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H e. (Subcat ` C ) )
17		eqid 2382	. . . . . . . . 9 |- ( Hom f ` C ) = ( Hom f ` C )
18	16, 17	subcssc 15246	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H C_cat ( Hom f ` C ) )
19		ssctr 15231	. . . . . . . 8 |- ( ( J C_cat H /\ H C_cat ( Hom f ` C ) ) -> J C_cat ( Hom f ` C ) )
20	15, 18, 19	syl2anc 659	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` C ) )
21	12	biimpa 482	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J C_cat ( Hom f ` D ) )
22	20, 21	2thd 240	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J C_cat ( Hom f ` C ) <-> J C_cat ( Hom f ` D ) ) )
23	16	adantr 463	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> H e. (Subcat ` C ) )
24	9	adantr 463	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> H Fn ( dom dom H X. dom dom H ) )
25	24	adantr 463	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> H Fn ( dom dom H X. dom dom H ) )
26		eqid 2382	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
27		eqidd 2383	. . . . . . . . . . . 12 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J = dom dom J )
28	15, 27	sscfn1 15223	. . . . . . . . . . 11 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> J Fn ( dom dom J X. dom dom J ) )
29	28, 24, 15	ssc1 15227	. . . . . . . . . 10 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom J C_ dom dom H )
30	29	sselda 3417	. . . . . . . . 9 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> x e. dom dom H )
31	4, 23, 25, 26, 30	subcid 15253	. . . . . . . 8 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( Id ` C ) ` x ) = ( ( Id ` D ) ` x ) )
32	31	eleq1d 2451	. . . . . . 7 |- ( ( ( H e. (Subcat ` C ) /\ J C_cat H ) /\ x e. dom dom J ) -> ( ( ( Id ` C ) ` x ) e. ( x J x ) <-> ( ( Id ` D ) ` x ) e. ( x J x ) ) )
33	32	ralbidva 2818	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( A. x e. dom dom J ( ( Id ` C ) ` x ) e. ( x J x ) <-> A. x e. dom dom J ( ( Id ` D ) ` x ) e. ( x J x ) ) )
34	4	oveq1i 6206	. . . . . . . 8 |- ( D |`cat J ) = ( ( C |`cat H ) |`cat J )
35	6	adantr 463	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> C e. Cat )
36		dmexg 6630	. . . . . . . . . . 11 |- ( H e. (Subcat ` C ) -> dom H e. _V )
37		dmexg 6630	. . . . . . . . . . 11 |- ( dom H e. _V -> dom dom H e. _V )
38	36, 37	syl 16	. . . . . . . . . 10 |- ( H e. (Subcat ` C ) -> dom dom H e. _V )
39	38	adantr 463	. . . . . . . . 9 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> dom dom H e. _V )
40	35, 24, 28, 39, 29	rescabs 15239	. . . . . . . 8 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat H ) |`cat J ) = ( C |`cat J ) )
41	34, 40	syl5req 2436	. . . . . . 7 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( C |`cat J ) = ( D |`cat J ) )
42	41	eleq1d 2451	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( C |`cat J ) e. Cat <-> ( D |`cat J ) e. Cat ) )
43	22, 33, 42	3anbi123d 1297	. . . . 5 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( ( J C_cat ( Hom f ` C ) /\ A. x e. dom dom J ( ( Id ` C ) ` x ) e. ( x J x ) /\ ( C |`cat J ) e. Cat ) <-> ( J C_cat ( Hom f ` D ) /\ A. x e. dom dom J ( ( Id ` D ) ` x ) e. ( x J x ) /\ ( D |`cat J ) e. Cat ) ) )
44		eqid 2382	. . . . . 6 |- ( C |`cat J ) = ( C |`cat J )
45	17, 26, 44, 35, 28	issubc3 15255	. . . . 5 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` C ) <-> ( J C_cat ( Hom f ` C ) /\ A. x e. dom dom J ( ( Id ` C ) ` x ) e. ( x J x ) /\ ( C |`cat J ) e. Cat ) ) )
46		eqid 2382	. . . . . 6 |- ( Id ` D ) = ( Id ` D )
47		eqid 2382	. . . . . 6 |- ( D |`cat J ) = ( D |`cat J )
48	4, 7	subccat 15254	. . . . . . 7 |- ( H e. (Subcat ` C ) -> D e. Cat )
49	48	adantr 463	. . . . . 6 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> D e. Cat )
50	2, 46, 47, 49, 28	issubc3 15255	. . . . 5 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` D ) <-> ( J C_cat ( Hom f ` D ) /\ A. x e. dom dom J ( ( Id ` D ) ` x ) e. ( x J x ) /\ ( D |`cat J ) e. Cat ) ) )
51	43, 45, 50	3bitr4rd 286	. . . 4 |- ( ( H e. (Subcat ` C ) /\ J C_cat H ) -> ( J e. (Subcat ` D ) <-> J e. (Subcat ` C ) ) )
52	51	pm5.32da 639	. . 3 |- ( H e. (Subcat ` C ) -> ( ( J C_cat H /\ J e. (Subcat ` D ) ) <-> ( J C_cat H /\ J e. (Subcat ` C ) ) ) )
53	14, 52	bitrd 253	. 2 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J C_cat H /\ J e. (Subcat ` C ) ) ) )
54		ancom 448	. 2 |- ( ( J C_cat H /\ J e. (Subcat ` C ) ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) )
55	53, 54	syl6bb 261	1 |- ( H e. (Subcat ` C ) -> ( J e. (Subcat ` D ) <-> ( J e. (Subcat ` C ) /\ J C_cat H ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1826   A.wral 2732   _Vcvv 3034   class class class wbr 4367   X. cxp 4911   dom cdm 4913   Fn wfn 5491   ` cfv 5496  (class class class)co 6196   Basecbs 14634   Catccat 15071   Idccid 15072   Hom <sub>f</sub> chomf 15073   C_<sub>cat</sub> cssc 15213   |`_cat cresc 15214  Subcatcsubc 15215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-hom 14726  df-cco 14727  df-cat 15075  df-cid 15076  df-homf 15077  df-ssc 15216  df-resc 15217  df-subc 15218

This theorem is referenced by:  fldhmsubc  33092  fldhmsubcALTV  33111