Theorem tcel 8247

Description: The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)

Hypothesis

Ref	Expression
tc2.1	|- A e. _V

Assertion

Ref	Expression
tcel	|- ( B e. A -> ( TC ` B ) C_ ( TC ` A ) )

Proof of Theorem tcel

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tcvalg 8240	. 2 |- ( B e. A -> ( TC ` B ) = |^| { x | ( B C_ x /\ Tr x ) } )
2		ssel 3412	. . . . . . . 8 |- ( A C_ x -> ( B e. A -> B e. x ) )
3		trss 4499	. . . . . . . . 9 |- ( Tr x -> ( B e. x -> B C_ x ) )
4	3	com12 31	. . . . . . . 8 |- ( B e. x -> ( Tr x -> B C_ x ) )
5	2, 4	syl6com 35	. . . . . . 7 |- ( B e. A -> ( A C_ x -> ( Tr x -> B C_ x ) ) )
6	5	impd 438	. . . . . 6 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> B C_ x ) )
7		simpr 468	. . . . . . 7 |- ( ( A C_ x /\ Tr x ) -> Tr x )
8	7	a1i 11	. . . . . 6 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> Tr x ) )
9	6, 8	jcad 542	. . . . 5 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> ( B C_ x /\ Tr x ) ) )
10	9	ss2abdv 3488	. . . 4 |- ( B e. A -> { x | ( A C_ x /\ Tr x ) } C_ { x | ( B C_ x /\ Tr x ) } )
11		intss 4247	. . . 4 |- ( { x | ( A C_ x /\ Tr x ) } C_ { x | ( B C_ x /\ Tr x ) } -> |^| { x | ( B C_ x /\ Tr x ) } C_ |^| { x | ( A C_ x /\ Tr x ) } )
12	10, 11	syl 17	. . 3 |- ( B e. A -> |^| { x | ( B C_ x /\ Tr x ) } C_ |^| { x | ( A C_ x /\ Tr x ) } )
13		tc2.1	. . . 4 |- A e. _V
14		tcvalg 8240	. . . 4 |- ( A e. _V -> ( TC ` A ) = |^| { x | ( A C_ x /\ Tr x ) } )
15	13, 14	ax-mp 5	. . 3 |- ( TC ` A ) = |^| { x | ( A C_ x /\ Tr x ) }
16	12, 15	syl6sseqr 3465	. 2 |- ( B e. A -> |^| { x | ( B C_ x /\ Tr x ) } C_ ( TC ` A ) )
17	1, 16	eqsstrd 3452	1 |- ( B e. A -> ( TC ` B ) C_ ( TC ` A ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   _Vcvv 3031   C_ wss 3390   |^|cint 4226   Tr wtr 4490   ` cfv 5589   TCctc 8238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-tc 8239

This theorem is referenced by:  tcrank  8373  hsmexlem4  8877