Theorem trclexi 37927

Description: The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)

Hypothesis

Ref	Expression
trclexi.1	|- A e. V

Assertion

Ref	Expression
trclexi	|- |^| { x | ( A C_ x /\ ( x o. x ) C_ x ) } e. _V

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem trclexi

Step	Hyp	Ref	Expression
1		ssun1 3776	. 2 |- A C_ ( A u. ( dom A X. ran A ) )
2		coundir 5637	. . . 4 |- ( ( A u. ( dom A X. ran A ) ) o. ( A u. ( dom A X. ran A ) ) ) = ( ( A o. ( A u. ( dom A X. ran A ) ) ) u. ( ( dom A X. ran A ) o. ( A u. ( dom A X. ran A ) ) ) )
3		coundi 5636	. . . . . 6 |- ( A o. ( A u. ( dom A X. ran A ) ) ) = ( ( A o. A ) u. ( A o. ( dom A X. ran A ) ) )
4		cossxp 5658	. . . . . . 7 |- ( A o. A ) C_ ( dom A X. ran A )
5		cossxp 5658	. . . . . . . 8 |- ( A o. ( dom A X. ran A ) ) C_ ( dom ( dom A X. ran A ) X. ran A )
6		dmxpss 5565	. . . . . . . . 9 |- dom ( dom A X. ran A ) C_ dom A
7		xpss1 5228	. . . . . . . . 9 |- ( dom ( dom A X. ran A ) C_ dom A -> ( dom ( dom A X. ran A ) X. ran A ) C_ ( dom A X. ran A ) )
8	6, 7	ax-mp 5	. . . . . . . 8 |- ( dom ( dom A X. ran A ) X. ran A ) C_ ( dom A X. ran A )
9	5, 8	sstri 3612	. . . . . . 7 |- ( A o. ( dom A X. ran A ) ) C_ ( dom A X. ran A )
10	4, 9	unssi 3788	. . . . . 6 |- ( ( A o. A ) u. ( A o. ( dom A X. ran A ) ) ) C_ ( dom A X. ran A )
11	3, 10	eqsstri 3635	. . . . 5 |- ( A o. ( A u. ( dom A X. ran A ) ) ) C_ ( dom A X. ran A )
12		coundi 5636	. . . . . 6 |- ( ( dom A X. ran A ) o. ( A u. ( dom A X. ran A ) ) ) = ( ( ( dom A X. ran A ) o. A ) u. ( ( dom A X. ran A ) o. ( dom A X. ran A ) ) )
13		cossxp 5658	. . . . . . . 8 |- ( ( dom A X. ran A ) o. A ) C_ ( dom A X. ran ( dom A X. ran A ) )
14		rnxpss 5566	. . . . . . . . 9 |- ran ( dom A X. ran A ) C_ ran A
15		xpss2 5229	. . . . . . . . 9 |- ( ran ( dom A X. ran A ) C_ ran A -> ( dom A X. ran ( dom A X. ran A ) ) C_ ( dom A X. ran A ) )
16	14, 15	ax-mp 5	. . . . . . . 8 |- ( dom A X. ran ( dom A X. ran A ) ) C_ ( dom A X. ran A )
17	13, 16	sstri 3612	. . . . . . 7 |- ( ( dom A X. ran A ) o. A ) C_ ( dom A X. ran A )
18		xptrrel 13719	. . . . . . 7 |- ( ( dom A X. ran A ) o. ( dom A X. ran A ) ) C_ ( dom A X. ran A )
19	17, 18	unssi 3788	. . . . . 6 |- ( ( ( dom A X. ran A ) o. A ) u. ( ( dom A X. ran A ) o. ( dom A X. ran A ) ) ) C_ ( dom A X. ran A )
20	12, 19	eqsstri 3635	. . . . 5 |- ( ( dom A X. ran A ) o. ( A u. ( dom A X. ran A ) ) ) C_ ( dom A X. ran A )
21	11, 20	unssi 3788	. . . 4 |- ( ( A o. ( A u. ( dom A X. ran A ) ) ) u. ( ( dom A X. ran A ) o. ( A u. ( dom A X. ran A ) ) ) ) C_ ( dom A X. ran A )
22	2, 21	eqsstri 3635	. . 3 |- ( ( A u. ( dom A X. ran A ) ) o. ( A u. ( dom A X. ran A ) ) ) C_ ( dom A X. ran A )
23		ssun2 3777	. . 3 |- ( dom A X. ran A ) C_ ( A u. ( dom A X. ran A ) )
24	22, 23	sstri 3612	. 2 |- ( ( A u. ( dom A X. ran A ) ) o. ( A u. ( dom A X. ran A ) ) ) C_ ( A u. ( dom A X. ran A ) )
25		trclexi.1	. . . . . 6 |- A e. V
26	25	elexi 3213	. . . . 5 |- A e. _V
27	26	dmex 7099	. . . . . 6 |- dom A e. _V
28	26	rnex 7100	. . . . . 6 |- ran A e. _V
29	27, 28	xpex 6962	. . . . 5 |- ( dom A X. ran A ) e. _V
30	26, 29	unex 6956	. . . 4 |- ( A u. ( dom A X. ran A ) ) e. _V
31		trcleq2lem 13730	. . . 4 |- ( x = ( A u. ( dom A X. ran A ) ) -> ( ( A C_ x /\ ( x o. x ) C_ x ) <-> ( A C_ ( A u. ( dom A X. ran A ) ) /\ ( ( A u. ( dom A X. ran A ) ) o. ( A u. ( dom A X. ran A ) ) ) C_ ( A u. ( dom A X. ran A ) ) ) ) )
32	30, 31	spcev 3300	. . 3 |- ( ( A C_ ( A u. ( dom A X. ran A ) ) /\ ( ( A u. ( dom A X. ran A ) ) o. ( A u. ( dom A X. ran A ) ) ) C_ ( A u. ( dom A X. ran A ) ) ) -> E. x ( A C_ x /\ ( x o. x ) C_ x ) )
33		intexab 4822	. . 3 |- ( E. x ( A C_ x /\ ( x o. x ) C_ x ) <-> |^| { x | ( A C_ x /\ ( x o. x ) C_ x ) } e. _V )
34	32, 33	sylib 208	. 2 |- ( ( A C_ ( A u. ( dom A X. ran A ) ) /\ ( ( A u. ( dom A X. ran A ) ) o. ( A u. ( dom A X. ran A ) ) ) C_ ( A u. ( dom A X. ran A ) ) ) -> |^| { x | ( A C_ x /\ ( x o. x ) C_ x ) } e. _V )
35	1, 24, 34	mp2an 708	1 |- |^| { x | ( A C_ x /\ ( x o. x ) C_ x ) } e. _V

Syntax hints:   /\ wa 384   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   u. cun 3572   C_ wss 3574   |^|cint 4475   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118

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-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-rab 2921  df-v 3202  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-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  dfrtrcl5  37936