Theorem tcel 8077

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 8070	. 2 |- ( B e. A -> ( TC ` B ) = |^| { x | ( B C_ x /\ Tr x ) } )
2		ssel 3459	. . . . . . . 8 |- ( A C_ x -> ( B e. A -> B e. x ) )
3		trss 4503	. . . . . . . . 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 431	. . . . . 6 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> B C_ x ) )
7		simpr 461	. . . . . . 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 533	. . . . 5 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> ( B C_ x /\ Tr x ) ) )
10	9	ss2abdv 3534	. . . 4 |- ( B e. A -> { x | ( A C_ x /\ Tr x ) } C_ { x | ( B C_ x /\ Tr x ) } )
11		intss 4258	. . . 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 16	. . 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 8070	. . . 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 3512	. 2 |- ( B e. A -> |^| { x | ( B C_ x /\ Tr x ) } C_ ( TC ` A ) )
17	1, 16	eqsstrd 3499	1 |- ( B e. A -> ( TC ` B ) C_ ( TC ` A ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   {cab 2439   _Vcvv 3078   C_ wss 3437   |^|cint 4237   Tr wtr 4494   ` cfv 5527   TCctc 8068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-tc 8069

This theorem is referenced by:  tcrank  8203  hsmexlem4  8710