Theorem itunitc1 9242

Description: Each union iterate is a member of the transitive closure. (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
itunitc1	|- ( ( U ` A ) ` B ) C_ ( TC ` A )

Distinct variable groups:   x, A, y   x, B, y

Allowed substitution hints:   U( x, y)

Proof of Theorem itunitc1

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( a = A -> ( U ` a ) = ( U ` A ) )
2	1	fveq1d 6193	. . . 4 |- ( a = A -> ( ( U ` a ) ` B ) = ( ( U ` A ) ` B ) )
3		fveq2 6191	. . . 4 |- ( a = A -> ( TC ` a ) = ( TC ` A ) )
4	2, 3	sseq12d 3634	. . 3 |- ( a = A -> ( ( ( U ` a ) ` B ) C_ ( TC ` a ) <-> ( ( U ` A ) ` B ) C_ ( TC ` A ) ) )
5		fveq2 6191	. . . . . 6 |- ( b = (/) -> ( ( U ` a ) ` b ) = ( ( U ` a ) ` (/) ) )
6	5	sseq1d 3632	. . . . 5 |- ( b = (/) -> ( ( ( U ` a ) ` b ) C_ ( TC ` a ) <-> ( ( U ` a ) ` (/) ) C_ ( TC ` a ) ) )
7		fveq2 6191	. . . . . 6 |- ( b = c -> ( ( U ` a ) ` b ) = ( ( U ` a ) ` c ) )
8	7	sseq1d 3632	. . . . 5 |- ( b = c -> ( ( ( U ` a ) ` b ) C_ ( TC ` a ) <-> ( ( U ` a ) ` c ) C_ ( TC ` a ) ) )
9		fveq2 6191	. . . . . 6 |- ( b = suc c -> ( ( U ` a ) ` b ) = ( ( U ` a ) ` suc c ) )
10	9	sseq1d 3632	. . . . 5 |- ( b = suc c -> ( ( ( U ` a ) ` b ) C_ ( TC ` a ) <-> ( ( U ` a ) ` suc c ) C_ ( TC ` a ) ) )
11		fveq2 6191	. . . . . 6 |- ( b = B -> ( ( U ` a ) ` b ) = ( ( U ` a ) ` B ) )
12	11	sseq1d 3632	. . . . 5 |- ( b = B -> ( ( ( U ` a ) ` b ) C_ ( TC ` a ) <-> ( ( U ` a ) ` B ) C_ ( TC ` a ) ) )
13		vex 3203	. . . . . 6 |- a e. _V
14		ituni.u	. . . . . . . 8 |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )
15	14	ituni0 9240	. . . . . . 7 |- ( a e. _V -> ( ( U ` a ) ` (/) ) = a )
16		tcid 8615	. . . . . . 7 |- ( a e. _V -> a C_ ( TC ` a ) )
17	15, 16	eqsstrd 3639	. . . . . 6 |- ( a e. _V -> ( ( U ` a ) ` (/) ) C_ ( TC ` a ) )
18	13, 17	ax-mp 5	. . . . 5 |- ( ( U ` a ) ` (/) ) C_ ( TC ` a )
19	14	itunisuc 9241	. . . . . . 7 |- ( ( U ` a ) ` suc c ) = U. ( ( U ` a ) ` c )
20		tctr 8616	. . . . . . . . . 10 |- Tr ( TC ` a )
21		pwtr 4921	. . . . . . . . . 10 |- ( Tr ( TC ` a ) <-> Tr ~P ( TC ` a ) )
22	20, 21	mpbi 220	. . . . . . . . 9 |- Tr ~P ( TC ` a )
23		trss 4761	. . . . . . . . 9 |- ( Tr ~P ( TC ` a ) -> ( ( ( U ` a ) ` c ) e. ~P ( TC ` a ) -> ( ( U ` a ) ` c ) C_ ~P ( TC ` a ) ) )
24	22, 23	ax-mp 5	. . . . . . . 8 |- ( ( ( U ` a ) ` c ) e. ~P ( TC ` a ) -> ( ( U ` a ) ` c ) C_ ~P ( TC ` a ) )
25		fvex 6201	. . . . . . . . 9 |- ( ( U ` a ) ` c ) e. _V
26	25	elpw 4164	. . . . . . . 8 |- ( ( ( U ` a ) ` c ) e. ~P ( TC ` a ) <-> ( ( U ` a ) ` c ) C_ ( TC ` a ) )
27		sspwuni 4611	. . . . . . . 8 |- ( ( ( U ` a ) ` c ) C_ ~P ( TC ` a ) <-> U. ( ( U ` a ) ` c ) C_ ( TC ` a ) )
28	24, 26, 27	3imtr3i 280	. . . . . . 7 |- ( ( ( U ` a ) ` c ) C_ ( TC ` a ) -> U. ( ( U ` a ) ` c ) C_ ( TC ` a ) )
29	19, 28	syl5eqss 3649	. . . . . 6 |- ( ( ( U ` a ) ` c ) C_ ( TC ` a ) -> ( ( U ` a ) ` suc c ) C_ ( TC ` a ) )
30	29	a1i 11	. . . . 5 |- ( c e. om -> ( ( ( U ` a ) ` c ) C_ ( TC ` a ) -> ( ( U ` a ) ` suc c ) C_ ( TC ` a ) ) )
31	6, 8, 10, 12, 18, 30	finds 7092	. . . 4 |- ( B e. om -> ( ( U ` a ) ` B ) C_ ( TC ` a ) )
32	14	itunifn 9239	. . . . . . . 8 |- ( a e. _V -> ( U ` a ) Fn om )
33		fndm 5990	. . . . . . . 8 |- ( ( U ` a ) Fn om -> dom ( U ` a ) = om )
34	13, 32, 33	mp2b 10	. . . . . . 7 |- dom ( U ` a ) = om
35	34	eleq2i 2693	. . . . . 6 |- ( B e. dom ( U ` a ) <-> B e. om )
36		ndmfv 6218	. . . . . 6 |- ( -. B e. dom ( U ` a ) -> ( ( U ` a ) ` B ) = (/) )
37	35, 36	sylnbir 321	. . . . 5 |- ( -. B e. om -> ( ( U ` a ) ` B ) = (/) )
38		0ss 3972	. . . . 5 |- (/) C_ ( TC ` a )
39	37, 38	syl6eqss 3655	. . . 4 |- ( -. B e. om -> ( ( U ` a ) ` B ) C_ ( TC ` a ) )
40	31, 39	pm2.61i 176	. . 3 |- ( ( U ` a ) ` B ) C_ ( TC ` a )
41	4, 40	vtoclg 3266	. 2 |- ( A e. _V -> ( ( U ` A ) ` B ) C_ ( TC ` A ) )
42		fvprc 6185	. . . . 5 |- ( -. A e. _V -> ( U ` A ) = (/) )
43	42	fveq1d 6193	. . . 4 |- ( -. A e. _V -> ( ( U ` A ) ` B ) = ( (/) ` B ) )
44		0fv 6227	. . . 4 |- ( (/) ` B ) = (/)
45	43, 44	syl6eq 2672	. . 3 |- ( -. A e. _V -> ( ( U ` A ) ` B ) = (/) )
46		0ss 3972	. . 3 |- (/) C_ ( TC ` A )
47	45, 46	syl6eqss 3655	. 2 |- ( -. A e. _V -> ( ( U ` A ) ` B ) C_ ( TC ` A ) )
48	41, 47	pm2.61i 176	1 |- ( ( U ` A ) ` B ) C_ ( TC ` A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   Tr wtr 4752   dom cdm 5114   |` 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:  itunitc  9243