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Theorem fullsubc 14049
 Description: The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
fullsubc.b
fullsubc.h f
fullsubc.c
fullsubc.s
Assertion
Ref Expression
fullsubc Subcat

Proof of Theorem fullsubc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullsubc.h . . . . 5 f
2 fullsubc.b . . . . 5
31, 2homffn 13921 . . . 4
4 fvex 5744 . . . . 5
52, 4eqeltri 2508 . . . 4
6 sscres 14025 . . . 4 cat
73, 5, 6mp2an 655 . . 3 cat
87a1i 11 . 2 cat
9 eqid 2438 . . . . . 6
10 eqid 2438 . . . . . 6
11 fullsubc.c . . . . . . 7
1211adantr 453 . . . . . 6
13 fullsubc.s . . . . . . 7
1413sselda 3350 . . . . . 6
152, 9, 10, 12, 14catidcl 13909 . . . . 5
16 simpr 449 . . . . . . 7
1716, 16ovresd 6216 . . . . . 6
181, 2, 9, 14, 14homfval 13920 . . . . . 6
1917, 18eqtrd 2470 . . . . 5
2015, 19eleqtrrd 2515 . . . 4
21 eqid 2438 . . . . . . . . . 10 comp comp
2212ad3antrrr 712 . . . . . . . . . 10
2314ad3antrrr 712 . . . . . . . . . 10
2413adantr 453 . . . . . . . . . . . . 13
2524sselda 3350 . . . . . . . . . . . 12
2625adantr 453 . . . . . . . . . . 11
2726adantr 453 . . . . . . . . . 10
2824adantr 453 . . . . . . . . . . . 12
2928sselda 3350 . . . . . . . . . . 11
3029adantr 453 . . . . . . . . . 10
31 simprl 734 . . . . . . . . . 10
32 simprr 735 . . . . . . . . . 10
332, 9, 21, 22, 23, 27, 30, 31, 32catcocl 13912 . . . . . . . . 9 comp
3416ad3antrrr 712 . . . . . . . . . . 11
35 simplr 733 . . . . . . . . . . 11
3634, 35ovresd 6216 . . . . . . . . . 10
371, 2, 9, 23, 30homfval 13920 . . . . . . . . . 10
3836, 37eqtrd 2470 . . . . . . . . 9
3933, 38eleqtrrd 2515 . . . . . . . 8 comp
4039ralrimivva 2800 . . . . . . 7 comp
41 simplr 733 . . . . . . . . . . 11
42 simpr 449 . . . . . . . . . . 11
4341, 42ovresd 6216 . . . . . . . . . 10
4414adantr 453 . . . . . . . . . . 11
451, 2, 9, 44, 25homfval 13920 . . . . . . . . . 10
4643, 45eqtrd 2470 . . . . . . . . 9
4746adantr 453 . . . . . . . 8
48 simplr 733 . . . . . . . . . . 11
49 simpr 449 . . . . . . . . . . 11
5048, 49ovresd 6216 . . . . . . . . . 10
511, 2, 9, 26, 29homfval 13920 . . . . . . . . . 10
5250, 51eqtrd 2470 . . . . . . . . 9
5352raleqdv 2912 . . . . . . . 8 comp comp
5447, 53raleqbidv 2918 . . . . . . 7 comp comp
5540, 54mpbird 225 . . . . . 6 comp
5655ralrimiva 2791 . . . . 5 comp
5756ralrimiva 2791 . . . 4 comp
5820, 57jca 520 . . 3 comp
5958ralrimiva 2791 . 2 comp
60 xpss12 4983 . . . . 5
6113, 13, 60syl2anc 644 . . . 4
62 fnssres 5560 . . . 4
633, 61, 62sylancr 646 . . 3
641, 10, 21, 11, 63issubc2 14038 . 2 Subcat cat comp
658, 59, 64mpbir2and 890 1 Subcat
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wral 2707  cvv 2958   wss 3322  cop 3819   class class class wbr 4214   cxp 4878   cres 4882   wfn 5451  cfv 5456  (class class class)co 6083  cbs 13471   chom 13542  compcco 13543  ccat 13891  ccid 13892   f chomf 13893   cat cssc 14009  Subcatcsubc 14011 This theorem is referenced by:  resscat  14051  funcres2c  14100  ressffth  14137  funcsetcres2  14250 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-pm 7023  df-ixp 7066  df-cat 13895  df-cid 13896  df-homf 13897  df-ssc 14012  df-subc 14014
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