Theorem onoviun 7440

Description: A variant of onovuni 7439 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Hypotheses

Ref	Expression
onovuni.1	|- ( Lim y -> ( A F y ) = U_ x e. y ( A F x ) )
onovuni.2	|- ( ( x e. On /\ y e. On /\ x C_ y ) -> ( A F x ) C_ ( A F y ) )

Assertion

Ref	Expression
onoviun	|- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( A F U_ z e. K L ) = U_ z e. K ( A F L ) )

Distinct variable groups:   x, y, z, A   x, F, y, z   x, K, y, z   x, L, y

Allowed substitution hints:   T( x, y, z)   L( z)

Proof of Theorem onoviun

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun3g 5378	. . . 4 |- ( A. z e. K L e. On -> U_ z e. K L = U. ran ( z e. K |-> L ) )
2	1	3ad2ant2 1083	. . 3 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> U_ z e. K L = U. ran ( z e. K |-> L ) )
3	2	oveq2d 6666	. 2 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( A F U_ z e. K L ) = ( A F U. ran ( z e. K |-> L ) ) )
4		simp1 1061	. . . 4 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> K e. T )
5		mptexg 6484	. . . 4 |- ( K e. T -> ( z e. K |-> L ) e. _V )
6		rnexg 7098	. . . 4 |- ( ( z e. K |-> L ) e. _V -> ran ( z e. K |-> L ) e. _V )
7	4, 5, 6	3syl 18	. . 3 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ran ( z e. K |-> L ) e. _V )
8		simp2 1062	. . . . 5 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> A. z e. K L e. On )
9		eqid 2622	. . . . . 6 |- ( z e. K |-> L ) = ( z e. K |-> L )
10	9	fmpt 6381	. . . . 5 |- ( A. z e. K L e. On <-> ( z e. K |-> L ) : K --> On )
11	8, 10	sylib 208	. . . 4 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( z e. K |-> L ) : K --> On )
12		frn 6053	. . . 4 |- ( ( z e. K |-> L ) : K --> On -> ran ( z e. K |-> L ) C_ On )
13	11, 12	syl 17	. . 3 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ran ( z e. K |-> L ) C_ On )
14		dmmptg 5632	. . . . . 6 |- ( A. z e. K L e. On -> dom ( z e. K |-> L ) = K )
15	14	3ad2ant2 1083	. . . . 5 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> dom ( z e. K |-> L ) = K )
16		simp3 1063	. . . . 5 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> K =/= (/) )
17	15, 16	eqnetrd 2861	. . . 4 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> dom ( z e. K |-> L ) =/= (/) )
18		dm0rn0 5342	. . . . 5 |- ( dom ( z e. K |-> L ) = (/) <-> ran ( z e. K |-> L ) = (/) )
19	18	necon3bii 2846	. . . 4 |- ( dom ( z e. K |-> L ) =/= (/) <-> ran ( z e. K |-> L ) =/= (/) )
20	17, 19	sylib 208	. . 3 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ran ( z e. K |-> L ) =/= (/) )
21		onovuni.1	. . . 4 |- ( Lim y -> ( A F y ) = U_ x e. y ( A F x ) )
22		onovuni.2	. . . 4 |- ( ( x e. On /\ y e. On /\ x C_ y ) -> ( A F x ) C_ ( A F y ) )
23	21, 22	onovuni 7439	. . 3 |- ( ( ran ( z e. K |-> L ) e. _V /\ ran ( z e. K |-> L ) C_ On /\ ran ( z e. K |-> L ) =/= (/) ) -> ( A F U. ran ( z e. K |-> L ) ) = U_ x e. ran ( z e. K |-> L ) ( A F x ) )
24	7, 13, 20, 23	syl3anc 1326	. 2 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( A F U. ran ( z e. K |-> L ) ) = U_ x e. ran ( z e. K |-> L ) ( A F x ) )
25		oveq2 6658	. . . . . . 7 |- ( x = L -> ( A F x ) = ( A F L ) )
26	25	eleq2d 2687	. . . . . 6 |- ( x = L -> ( w e. ( A F x ) <-> w e. ( A F L ) ) )
27	9, 26	rexrnmpt 6369	. . . . 5 |- ( A. z e. K L e. On -> ( E. x e. ran ( z e. K |-> L ) w e. ( A F x ) <-> E. z e. K w e. ( A F L ) ) )
28	27	3ad2ant2 1083	. . . 4 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( E. x e. ran ( z e. K |-> L ) w e. ( A F x ) <-> E. z e. K w e. ( A F L ) ) )
29		eliun 4524	. . . 4 |- ( w e. U_ x e. ran ( z e. K |-> L ) ( A F x ) <-> E. x e. ran ( z e. K |-> L ) w e. ( A F x ) )
30		eliun 4524	. . . 4 |- ( w e. U_ z e. K ( A F L ) <-> E. z e. K w e. ( A F L ) )
31	28, 29, 30	3bitr4g 303	. . 3 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( w e. U_ x e. ran ( z e. K |-> L ) ( A F x ) <-> w e. U_ z e. K ( A F L ) ) )
32	31	eqrdv 2620	. 2 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> U_ x e. ran ( z e. K |-> L ) ( A F x ) = U_ z e. K ( A F L ) )
33	3, 24, 32	3eqtrd 2660	1 |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) ) -> ( A F U_ z e. K L ) = U_ z e. K ( A F L ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Oncon0 5723   Lim wlim 5724   -->wf 5884  (class class class)co 6650

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

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-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-ord 5726  df-on 5727  df-lim 5728  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-ov 6653

This theorem is referenced by:  oeoalem  7676  oeoelem  7678