Theorem tcel 8621

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 8614	. 2 |- ( B e. A -> ( TC ` B ) = |^| { x | ( B C_ x /\ Tr x ) } )
2		ssel 3597	. . . . . . . 8 |- ( A C_ x -> ( B e. A -> B e. x ) )
3		trss 4761	. . . . . . . . 9 |- ( Tr x -> ( B e. x -> B C_ x ) )
4	3	com12 32	. . . . . . . 8 |- ( B e. x -> ( Tr x -> B C_ x ) )
5	2, 4	syl6com 37	. . . . . . 7 |- ( B e. A -> ( A C_ x -> ( Tr x -> B C_ x ) ) )
6	5	impd 447	. . . . . 6 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> B C_ x ) )
7		simpr 477	. . . . . . 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 555	. . . . 5 |- ( B e. A -> ( ( A C_ x /\ Tr x ) -> ( B C_ x /\ Tr x ) ) )
10	9	ss2abdv 3675	. . . 4 |- ( B e. A -> { x | ( A C_ x /\ Tr x ) } C_ { x | ( B C_ x /\ Tr x ) } )
11		intss 4498	. . . 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 8614	. . . 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 3652	. 2 |- ( B e. A -> |^| { x | ( B C_ x /\ Tr x ) } C_ ( TC ` A ) )
17	1, 16	eqsstrd 3639	1 |- ( B e. A -> ( TC ` B ) C_ ( TC ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   |^|cint 4475   Tr wtr 4752   ` cfv 5888   TCctc 8612

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  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-tc 8613

This theorem is referenced by:  tcrank  8747  hsmexlem4  9251