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Theorem ssctr 14030
 Description: The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
ssctr cat cat cat

Proof of Theorem ssctr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 445 . . . . 5 cat cat cat
2 eqidd 2439 . . . . 5 cat cat
31, 2sscfn1 14022 . . . 4 cat cat
4 eqidd 2439 . . . . 5 cat cat
51, 4sscfn2 14023 . . . 4 cat cat
63, 5, 1ssc1 14026 . . 3 cat cat
7 simpr 449 . . . . 5 cat cat cat
8 eqidd 2439 . . . . 5 cat cat
97, 8sscfn2 14023 . . . 4 cat cat
105, 9, 7ssc1 14026 . . 3 cat cat
116, 10sstrd 3360 . 2 cat cat
123adantr 453 . . . . 5 cat cat
131adantr 453 . . . . 5 cat cat cat
14 simprl 734 . . . . 5 cat cat
15 simprr 735 . . . . 5 cat cat
1612, 13, 14, 15ssc2 14027 . . . 4 cat cat
175adantr 453 . . . . 5 cat cat
187adantr 453 . . . . 5 cat cat cat
196adantr 453 . . . . . 6 cat cat
2019, 14sseldd 3351 . . . . 5 cat cat
2119, 15sseldd 3351 . . . . 5 cat cat
2217, 18, 20, 21ssc2 14027 . . . 4 cat cat
2316, 22sstrd 3360 . . 3 cat cat
2423ralrimivva 2800 . 2 cat cat
25 sscrel 14018 . . . . . 6 cat
2625brrelex2i 4922 . . . . 5 cat
2726adantl 454 . . . 4 cat cat
28 dmexg 5133 . . . 4
29 dmexg 5133 . . . 4
3027, 28, 293syl 19 . . 3 cat cat
313, 9, 30isssc 14025 . 2 cat cat cat
3211, 24, 31mpbir2and 890 1 cat cat cat
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wcel 1726  wral 2707  cvv 2958   wss 3322   class class class wbr 4215   cxp 4879   cdm 4881   wfn 5452  (class class class)co 6084   cat cssc 14012 This theorem is referenced by:  subsubc  14055 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 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-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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-ixp 7067  df-ssc 14015
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