Theorem tcel 8167

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 8160	. 2 |- ( B e. A -> ( TC ` B ) = |^| { x | ( B C_ x /\ Tr x ) } )
2		ssel 3483	. . . . . . . 8 |- ( A C_ x -> ( B e. A -> B e. x ) )
3		trss 4541	. . . . . . . . 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 429	. . . . . 6 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> B C_ x ) )
7		simpr 459	. . . . . . 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 531	. . . . 5 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> ( B C_ x /\ Tr x ) ) )
10	9	ss2abdv 3559	. . . 4 |- ( B e. A -> { x | ( A C_ x /\ Tr x ) } C_ { x | ( B C_ x /\ Tr x ) } )
11		intss 4292	. . . 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 8160	. . . 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 3536	. 2 |- ( B e. A -> |^| { x | ( B C_ x /\ Tr x ) } C_ ( TC ` A ) )
17	1, 16	eqsstrd 3523	1 |- ( B e. A -> ( TC ` B ) C_ ( TC ` A ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   {cab 2439   _Vcvv 3106   C_ wss 3461   |^|cint 4271   Tr wtr 4532   ` cfv 5570   TCctc 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-tc 8159

This theorem is referenced by:  tcrank  8293  hsmexlem4  8800