Theorem ssctr 16485

Description: The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
ssctr	|- ( ( A C_cat B /\ B C_cat C ) -> A C_cat C )

Proof of Theorem ssctr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( A C_cat B /\ B C_cat C ) -> A C_cat B )
2		eqidd 2623	. . . . 5 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom A = dom dom A )
3	1, 2	sscfn1 16477	. . . 4 |- ( ( A C_cat B /\ B C_cat C ) -> A Fn ( dom dom A X. dom dom A ) )
4		eqidd 2623	. . . . 5 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom B = dom dom B )
5	1, 4	sscfn2 16478	. . . 4 |- ( ( A C_cat B /\ B C_cat C ) -> B Fn ( dom dom B X. dom dom B ) )
6	3, 5, 1	ssc1 16481	. . 3 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom A C_ dom dom B )
7		simpr 477	. . . . 5 |- ( ( A C_cat B /\ B C_cat C ) -> B C_cat C )
8		eqidd 2623	. . . . 5 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom C = dom dom C )
9	7, 8	sscfn2 16478	. . . 4 |- ( ( A C_cat B /\ B C_cat C ) -> C Fn ( dom dom C X. dom dom C ) )
10	5, 9, 7	ssc1 16481	. . 3 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom B C_ dom dom C )
11	6, 10	sstrd 3613	. 2 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom A C_ dom dom C )
12	3	adantr 481	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> A Fn ( dom dom A X. dom dom A ) )
13	1	adantr 481	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> A C_cat B )
14		simprl 794	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> x e. dom dom A )
15		simprr 796	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> y e. dom dom A )
16	12, 13, 14, 15	ssc2 16482	. . . 4 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> ( x A y ) C_ ( x B y ) )
17	5	adantr 481	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> B Fn ( dom dom B X. dom dom B ) )
18	7	adantr 481	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> B C_cat C )
19	6	adantr 481	. . . . . 6 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> dom dom A C_ dom dom B )
20	19, 14	sseldd 3604	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> x e. dom dom B )
21	19, 15	sseldd 3604	. . . . 5 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> y e. dom dom B )
22	17, 18, 20, 21	ssc2 16482	. . . 4 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> ( x B y ) C_ ( x C y ) )
23	16, 22	sstrd 3613	. . 3 |- ( ( ( A C_cat B /\ B C_cat C ) /\ ( x e. dom dom A /\ y e. dom dom A ) ) -> ( x A y ) C_ ( x C y ) )
24	23	ralrimivva 2971	. 2 |- ( ( A C_cat B /\ B C_cat C ) -> A. x e. dom dom A A. y e. dom dom A ( x A y ) C_ ( x C y ) )
25		sscrel 16473	. . . . . 6 |- Rel C_cat
26	25	brrelex2i 5159	. . . . 5 |- ( B C_cat C -> C e. _V )
27	26	adantl 482	. . . 4 |- ( ( A C_cat B /\ B C_cat C ) -> C e. _V )
28		dmexg 7097	. . . 4 |- ( C e. _V -> dom C e. _V )
29		dmexg 7097	. . . 4 |- ( dom C e. _V -> dom dom C e. _V )
30	27, 28, 29	3syl 18	. . 3 |- ( ( A C_cat B /\ B C_cat C ) -> dom dom C e. _V )
31	3, 9, 30	isssc 16480	. 2 |- ( ( A C_cat B /\ B C_cat C ) -> ( A C_cat C <-> ( dom dom A C_ dom dom C /\ A. x e. dom dom A A. y e. dom dom A ( x A y ) C_ ( x C y ) ) ) )
32	11, 24, 31	mpbir2and 957	1 |- ( ( A C_cat B /\ B C_cat C ) -> A C_cat C )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Fn wfn 5883  (class class class)co 6650   C__cat cssc 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-ixp 7909  df-ssc 16470

This theorem is referenced by:  subsubc  16513