Theorem sscfn1 16477

Description: The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
sscfn1.1	|- ( ph -> H C_cat J )
sscfn1.2	|- ( ph -> S = dom dom H )

Assertion

Ref	Expression
sscfn1	|- ( ph -> H Fn ( S X. S ) )

Proof of Theorem sscfn1

Dummy variables t s x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sscfn1.1	. . 3 |- ( ph -> H C_cat J )
2		brssc 16474	. . 3 |- ( H C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )
3	1, 2	sylib 208	. 2 |- ( ph -> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )
4		ixpfn 7914	. . . . . 6 |- ( H e. X_ x e. ( s X. s ) ~P ( J ` x ) -> H Fn ( s X. s ) )
5		simpr 477	. . . . . . . 8 |- ( ( ph /\ H Fn ( s X. s ) ) -> H Fn ( s X. s ) )
6		sscfn1.2	. . . . . . . . . . . 12 |- ( ph -> S = dom dom H )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ H Fn ( s X. s ) ) -> S = dom dom H )
8		fndm 5990	. . . . . . . . . . . . . 14 |- ( H Fn ( s X. s ) -> dom H = ( s X. s ) )
9	8	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ H Fn ( s X. s ) ) -> dom H = ( s X. s ) )
10	9	dmeqd 5326	. . . . . . . . . . . 12 |- ( ( ph /\ H Fn ( s X. s ) ) -> dom dom H = dom ( s X. s ) )
11		dmxpid 5345	. . . . . . . . . . . 12 |- dom ( s X. s ) = s
12	10, 11	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ph /\ H Fn ( s X. s ) ) -> dom dom H = s )
13	7, 12	eqtr2d 2657	. . . . . . . . . 10 |- ( ( ph /\ H Fn ( s X. s ) ) -> s = S )
14	13	sqxpeqd 5141	. . . . . . . . 9 |- ( ( ph /\ H Fn ( s X. s ) ) -> ( s X. s ) = ( S X. S ) )
15	14	fneq2d 5982	. . . . . . . 8 |- ( ( ph /\ H Fn ( s X. s ) ) -> ( H Fn ( s X. s ) <-> H Fn ( S X. S ) ) )
16	5, 15	mpbid 222	. . . . . . 7 |- ( ( ph /\ H Fn ( s X. s ) ) -> H Fn ( S X. S ) )
17	16	ex 450	. . . . . 6 |- ( ph -> ( H Fn ( s X. s ) -> H Fn ( S X. S ) ) )
18	4, 17	syl5 34	. . . . 5 |- ( ph -> ( H e. X_ x e. ( s X. s ) ~P ( J ` x ) -> H Fn ( S X. S ) ) )
19	18	rexlimdvw 3034	. . . 4 |- ( ph -> ( E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) -> H Fn ( S X. S ) ) )
20	19	adantld 483	. . 3 |- ( ph -> ( ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> H Fn ( S X. S ) ) )
21	20	exlimdv 1861	. 2 |- ( ph -> ( E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> H Fn ( S X. S ) ) )
22	3, 21	mpd 15	1 |- ( ph -> H Fn ( S X. S ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Fn wfn 5883   ` cfv 5888   X_cixp 7908   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-ixp 7909  df-ssc 16470

This theorem is referenced by:  ssctr  16485  ssceq  16486  subcfn  16501  subsubc  16513