Theorem ituniiun 9244

Description: Unwrap an iterated union from the "other end". (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
ituniiun	|- ( A e. V -> ( ( U ` A ) ` suc B ) = U_ a e. A ( ( U ` a ) ` B ) )

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

Allowed substitution hints:   U( x, y)   V( x, y, a)

Proof of Theorem ituniiun

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( b = A -> ( U ` b ) = ( U ` A ) )
2	1	fveq1d 6193	. . 3 |- ( b = A -> ( ( U ` b ) ` suc B ) = ( ( U ` A ) ` suc B ) )
3		iuneq1 4534	. . 3 |- ( b = A -> U_ a e. b ( ( U ` a ) ` B ) = U_ a e. A ( ( U ` a ) ` B ) )
4	2, 3	eqeq12d 2637	. 2 |- ( b = A -> ( ( ( U ` b ) ` suc B ) = U_ a e. b ( ( U ` a ) ` B ) <-> ( ( U ` A ) ` suc B ) = U_ a e. A ( ( U ` a ) ` B ) ) )
5		suceq 5790	. . . . . 6 |- ( d = (/) -> suc d = suc (/) )
6	5	fveq2d 6195	. . . . 5 |- ( d = (/) -> ( ( U ` b ) ` suc d ) = ( ( U ` b ) ` suc (/) ) )
7		fveq2 6191	. . . . . 6 |- ( d = (/) -> ( ( U ` a ) ` d ) = ( ( U ` a ) ` (/) ) )
8	7	iuneq2d 4547	. . . . 5 |- ( d = (/) -> U_ a e. b ( ( U ` a ) ` d ) = U_ a e. b ( ( U ` a ) ` (/) ) )
9	6, 8	eqeq12d 2637	. . . 4 |- ( d = (/) -> ( ( ( U ` b ) ` suc d ) = U_ a e. b ( ( U ` a ) ` d ) <-> ( ( U ` b ) ` suc (/) ) = U_ a e. b ( ( U ` a ) ` (/) ) ) )
10		suceq 5790	. . . . . 6 |- ( d = c -> suc d = suc c )
11	10	fveq2d 6195	. . . . 5 |- ( d = c -> ( ( U ` b ) ` suc d ) = ( ( U ` b ) ` suc c ) )
12		fveq2 6191	. . . . . 6 |- ( d = c -> ( ( U ` a ) ` d ) = ( ( U ` a ) ` c ) )
13	12	iuneq2d 4547	. . . . 5 |- ( d = c -> U_ a e. b ( ( U ` a ) ` d ) = U_ a e. b ( ( U ` a ) ` c ) )
14	11, 13	eqeq12d 2637	. . . 4 |- ( d = c -> ( ( ( U ` b ) ` suc d ) = U_ a e. b ( ( U ` a ) ` d ) <-> ( ( U ` b ) ` suc c ) = U_ a e. b ( ( U ` a ) ` c ) ) )
15		suceq 5790	. . . . . 6 |- ( d = suc c -> suc d = suc suc c )
16	15	fveq2d 6195	. . . . 5 |- ( d = suc c -> ( ( U ` b ) ` suc d ) = ( ( U ` b ) ` suc suc c ) )
17		fveq2 6191	. . . . . 6 |- ( d = suc c -> ( ( U ` a ) ` d ) = ( ( U ` a ) ` suc c ) )
18	17	iuneq2d 4547	. . . . 5 |- ( d = suc c -> U_ a e. b ( ( U ` a ) ` d ) = U_ a e. b ( ( U ` a ) ` suc c ) )
19	16, 18	eqeq12d 2637	. . . 4 |- ( d = suc c -> ( ( ( U ` b ) ` suc d ) = U_ a e. b ( ( U ` a ) ` d ) <-> ( ( U ` b ) ` suc suc c ) = U_ a e. b ( ( U ` a ) ` suc c ) ) )
20		suceq 5790	. . . . . 6 |- ( d = B -> suc d = suc B )
21	20	fveq2d 6195	. . . . 5 |- ( d = B -> ( ( U ` b ) ` suc d ) = ( ( U ` b ) ` suc B ) )
22		fveq2 6191	. . . . . 6 |- ( d = B -> ( ( U ` a ) ` d ) = ( ( U ` a ) ` B ) )
23	22	iuneq2d 4547	. . . . 5 |- ( d = B -> U_ a e. b ( ( U ` a ) ` d ) = U_ a e. b ( ( U ` a ) ` B ) )
24	21, 23	eqeq12d 2637	. . . 4 |- ( d = B -> ( ( ( U ` b ) ` suc d ) = U_ a e. b ( ( U ` a ) ` d ) <-> ( ( U ` b ) ` suc B ) = U_ a e. b ( ( U ` a ) ` B ) ) )
25		uniiun 4573	. . . . 5 |- U. b = U_ a e. b a
26		ituni.u	. . . . . . 7 |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )
27	26	itunisuc 9241	. . . . . 6 |- ( ( U ` b ) ` suc (/) ) = U. ( ( U ` b ) ` (/) )
28		vex 3203	. . . . . . . 8 |- b e. _V
29	26	ituni0 9240	. . . . . . . 8 |- ( b e. _V -> ( ( U ` b ) ` (/) ) = b )
30	28, 29	ax-mp 5	. . . . . . 7 |- ( ( U ` b ) ` (/) ) = b
31	30	unieqi 4445	. . . . . 6 |- U. ( ( U ` b ) ` (/) ) = U. b
32	27, 31	eqtri 2644	. . . . 5 |- ( ( U ` b ) ` suc (/) ) = U. b
33	26	ituni0 9240	. . . . . 6 |- ( a e. b -> ( ( U ` a ) ` (/) ) = a )
34	33	iuneq2i 4539	. . . . 5 |- U_ a e. b ( ( U ` a ) ` (/) ) = U_ a e. b a
35	25, 32, 34	3eqtr4i 2654	. . . 4 |- ( ( U ` b ) ` suc (/) ) = U_ a e. b ( ( U ` a ) ` (/) )
36	26	itunisuc 9241	. . . . . 6 |- ( ( U ` b ) ` suc suc c ) = U. ( ( U ` b ) ` suc c )
37		unieq 4444	. . . . . . 7 |- ( ( ( U ` b ) ` suc c ) = U_ a e. b ( ( U ` a ) ` c ) -> U. ( ( U ` b ) ` suc c ) = U. U_ a e. b ( ( U ` a ) ` c ) )
38	26	itunisuc 9241	. . . . . . . . . 10 |- ( ( U ` a ) ` suc c ) = U. ( ( U ` a ) ` c )
39	38	a1i 11	. . . . . . . . 9 |- ( a e. b -> ( ( U ` a ) ` suc c ) = U. ( ( U ` a ) ` c ) )
40	39	iuneq2i 4539	. . . . . . . 8 |- U_ a e. b ( ( U ` a ) ` suc c ) = U_ a e. b U. ( ( U ` a ) ` c )
41		iuncom4 4528	. . . . . . . 8 |- U_ a e. b U. ( ( U ` a ) ` c ) = U. U_ a e. b ( ( U ` a ) ` c )
42	40, 41	eqtr2i 2645	. . . . . . 7 |- U. U_ a e. b ( ( U ` a ) ` c ) = U_ a e. b ( ( U ` a ) ` suc c )
43	37, 42	syl6eq 2672	. . . . . 6 |- ( ( ( U ` b ) ` suc c ) = U_ a e. b ( ( U ` a ) ` c ) -> U. ( ( U ` b ) ` suc c ) = U_ a e. b ( ( U ` a ) ` suc c ) )
44	36, 43	syl5eq 2668	. . . . 5 |- ( ( ( U ` b ) ` suc c ) = U_ a e. b ( ( U ` a ) ` c ) -> ( ( U ` b ) ` suc suc c ) = U_ a e. b ( ( U ` a ) ` suc c ) )
45	44	a1i 11	. . . 4 |- ( c e. om -> ( ( ( U ` b ) ` suc c ) = U_ a e. b ( ( U ` a ) ` c ) -> ( ( U ` b ) ` suc suc c ) = U_ a e. b ( ( U ` a ) ` suc c ) ) )
46	9, 14, 19, 24, 35, 45	finds 7092	. . 3 |- ( B e. om -> ( ( U ` b ) ` suc B ) = U_ a e. b ( ( U ` a ) ` B ) )
47		iun0 4576	. . . . 5 |- U_ a e. b (/) = (/)
48	47	eqcomi 2631	. . . 4 |- (/) = U_ a e. b (/)
49		peano2b 7081	. . . . . 6 |- ( B e. om <-> suc B e. om )
50	26	itunifn 9239	. . . . . . . 8 |- ( b e. _V -> ( U ` b ) Fn om )
51		fndm 5990	. . . . . . . 8 |- ( ( U ` b ) Fn om -> dom ( U ` b ) = om )
52	28, 50, 51	mp2b 10	. . . . . . 7 |- dom ( U ` b ) = om
53	52	eleq2i 2693	. . . . . 6 |- ( suc B e. dom ( U ` b ) <-> suc B e. om )
54	49, 53	bitr4i 267	. . . . 5 |- ( B e. om <-> suc B e. dom ( U ` b ) )
55		ndmfv 6218	. . . . 5 |- ( -. suc B e. dom ( U ` b ) -> ( ( U ` b ) ` suc B ) = (/) )
56	54, 55	sylnbi 320	. . . 4 |- ( -. B e. om -> ( ( U ` b ) ` suc B ) = (/) )
57		vex 3203	. . . . . . . 8 |- a e. _V
58	26	itunifn 9239	. . . . . . . 8 |- ( a e. _V -> ( U ` a ) Fn om )
59		fndm 5990	. . . . . . . 8 |- ( ( U ` a ) Fn om -> dom ( U ` a ) = om )
60	57, 58, 59	mp2b 10	. . . . . . 7 |- dom ( U ` a ) = om
61	60	eleq2i 2693	. . . . . 6 |- ( B e. dom ( U ` a ) <-> B e. om )
62		ndmfv 6218	. . . . . 6 |- ( -. B e. dom ( U ` a ) -> ( ( U ` a ) ` B ) = (/) )
63	61, 62	sylnbir 321	. . . . 5 |- ( -. B e. om -> ( ( U ` a ) ` B ) = (/) )
64	63	iuneq2d 4547	. . . 4 |- ( -. B e. om -> U_ a e. b ( ( U ` a ) ` B ) = U_ a e. b (/) )
65	48, 56, 64	3eqtr4a 2682	. . 3 |- ( -. B e. om -> ( ( U ` b ) ` suc B ) = U_ a e. b ( ( U ` a ) ` B ) )
66	46, 65	pm2.61i 176	. 2 |- ( ( U ` b ) ` suc B ) = U_ a e. b ( ( U ` a ) ` B )
67	4, 66	vtoclg 3266	1 |- ( A e. V -> ( ( U ` A ) ` suc B ) = U_ a e. A ( ( U ` a ) ` B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   dom cdm 5114   |` cres 5116   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   reccrdg 7505

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-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

This theorem is referenced by:  hsmexlem4  9251