Theorem itunitc 9243

Description: The union of all union iterates creates the transitive closure; compare trcl 8604. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Hypothesis

Ref	Expression
ituni.u	|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )

Assertion

Ref	Expression
itunitc	|- ( TC ` A ) = U. ran ( U ` A )

Distinct variable group:   x, A, y

Allowed substitution hints:   U( x, y)

Proof of Theorem itunitc

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( a = A -> ( TC ` a ) = ( TC ` A ) )
2		fveq2 6191	. . . . . 6 |- ( a = A -> ( U ` a ) = ( U ` A ) )
3	2	rneqd 5353	. . . . 5 |- ( a = A -> ran ( U ` a ) = ran ( U ` A ) )
4	3	unieqd 4446	. . . 4 |- ( a = A -> U. ran ( U ` a ) = U. ran ( U ` A ) )
5	1, 4	eqeq12d 2637	. . 3 |- ( a = A -> ( ( TC ` a ) = U. ran ( U ` a ) <-> ( TC ` A ) = U. ran ( U ` A ) ) )
6		vex 3203	. . . . . . 7 |- a e. _V
7		ituni.u	. . . . . . . 8 |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )
8	7	ituni0 9240	. . . . . . 7 |- ( a e. _V -> ( ( U ` a ) ` (/) ) = a )
9	6, 8	ax-mp 5	. . . . . 6 |- ( ( U ` a ) ` (/) ) = a
10		fvssunirn 6217	. . . . . 6 |- ( ( U ` a ) ` (/) ) C_ U. ran ( U ` a )
11	9, 10	eqsstr3i 3636	. . . . 5 |- a C_ U. ran ( U ` a )
12		dftr3 4756	. . . . . 6 |- ( Tr U. ran ( U ` a ) <-> A. b e. U. ran ( U ` a ) b C_ U. ran ( U ` a ) )
13	7	itunifn 9239	. . . . . . . 8 |- ( a e. _V -> ( U ` a ) Fn om )
14		fnunirn 6511	. . . . . . . 8 |- ( ( U ` a ) Fn om -> ( b e. U. ran ( U ` a ) <-> E. c e. om b e. ( ( U ` a ) ` c ) ) )
15	6, 13, 14	mp2b 10	. . . . . . 7 |- ( b e. U. ran ( U ` a ) <-> E. c e. om b e. ( ( U ` a ) ` c ) )
16		elssuni 4467	. . . . . . . . 9 |- ( b e. ( ( U ` a ) ` c ) -> b C_ U. ( ( U ` a ) ` c ) )
17	7	itunisuc 9241	. . . . . . . . . 10 |- ( ( U ` a ) ` suc c ) = U. ( ( U ` a ) ` c )
18		fvssunirn 6217	. . . . . . . . . 10 |- ( ( U ` a ) ` suc c ) C_ U. ran ( U ` a )
19	17, 18	eqsstr3i 3636	. . . . . . . . 9 |- U. ( ( U ` a ) ` c ) C_ U. ran ( U ` a )
20	16, 19	syl6ss 3615	. . . . . . . 8 |- ( b e. ( ( U ` a ) ` c ) -> b C_ U. ran ( U ` a ) )
21	20	rexlimivw 3029	. . . . . . 7 |- ( E. c e. om b e. ( ( U ` a ) ` c ) -> b C_ U. ran ( U ` a ) )
22	15, 21	sylbi 207	. . . . . 6 |- ( b e. U. ran ( U ` a ) -> b C_ U. ran ( U ` a ) )
23	12, 22	mprgbir 2927	. . . . 5 |- Tr U. ran ( U ` a )
24		tcmin 8617	. . . . . 6 |- ( a e. _V -> ( ( a C_ U. ran ( U ` a ) /\ Tr U. ran ( U ` a ) ) -> ( TC ` a ) C_ U. ran ( U ` a ) ) )
25	6, 24	ax-mp 5	. . . . 5 |- ( ( a C_ U. ran ( U ` a ) /\ Tr U. ran ( U ` a ) ) -> ( TC ` a ) C_ U. ran ( U ` a ) )
26	11, 23, 25	mp2an 708	. . . 4 |- ( TC ` a ) C_ U. ran ( U ` a )
27		unissb 4469	. . . . 5 |- ( U. ran ( U ` a ) C_ ( TC ` a ) <-> A. b e. ran ( U ` a ) b C_ ( TC ` a ) )
28		fvelrnb 6243	. . . . . . 7 |- ( ( U ` a ) Fn om -> ( b e. ran ( U ` a ) <-> E. c e. om ( ( U ` a ) ` c ) = b ) )
29	6, 13, 28	mp2b 10	. . . . . 6 |- ( b e. ran ( U ` a ) <-> E. c e. om ( ( U ` a ) ` c ) = b )
30	7	itunitc1 9242	. . . . . . . . 9 |- ( ( U ` a ) ` c ) C_ ( TC ` a )
31	30	a1i 11	. . . . . . . 8 |- ( c e. om -> ( ( U ` a ) ` c ) C_ ( TC ` a ) )
32		sseq1 3626	. . . . . . . 8 |- ( ( ( U ` a ) ` c ) = b -> ( ( ( U ` a ) ` c ) C_ ( TC ` a ) <-> b C_ ( TC ` a ) ) )
33	31, 32	syl5ibcom 235	. . . . . . 7 |- ( c e. om -> ( ( ( U ` a ) ` c ) = b -> b C_ ( TC ` a ) ) )
34	33	rexlimiv 3027	. . . . . 6 |- ( E. c e. om ( ( U ` a ) ` c ) = b -> b C_ ( TC ` a ) )
35	29, 34	sylbi 207	. . . . 5 |- ( b e. ran ( U ` a ) -> b C_ ( TC ` a ) )
36	27, 35	mprgbir 2927	. . . 4 |- U. ran ( U ` a ) C_ ( TC ` a )
37	26, 36	eqssi 3619	. . 3 |- ( TC ` a ) = U. ran ( U ` a )
38	5, 37	vtoclg 3266	. 2 |- ( A e. _V -> ( TC ` A ) = U. ran ( U ` A ) )
39		rn0 5377	. . . . 5 |- ran (/) = (/)
40	39	unieqi 4445	. . . 4 |- U. ran (/) = U. (/)
41		uni0 4465	. . . 4 |- U. (/) = (/)
42	40, 41	eqtr2i 2645	. . 3 |- (/) = U. ran (/)
43		fvprc 6185	. . 3 |- ( -. A e. _V -> ( TC ` A ) = (/) )
44		fvprc 6185	. . . . 5 |- ( -. A e. _V -> ( U ` A ) = (/) )
45	44	rneqd 5353	. . . 4 |- ( -. A e. _V -> ran ( U ` A ) = ran (/) )
46	45	unieqd 4446	. . 3 |- ( -. A e. _V -> U. ran ( U ` A ) = U. ran (/) )
47	42, 43, 46	3eqtr4a 2682	. 2 |- ( -. A e. _V -> ( TC ` A ) = U. ran ( U ` A ) )
48	38, 47	pm2.61i 176	1 |- ( TC ` A ) = U. ran ( U ` A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   |-> cmpt 4729   Tr wtr 4752   ran crn 5115   |` cres 5116   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   reccrdg 7505   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:  hsmexlem5  9252