Theorem onovuni 7439

Description: A variant of onfununi 7438 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

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
onovuni	|- ( ( S e. T /\ S C_ On /\ S =/= (/) ) -> ( A F U. S ) = U_ x e. S ( A F x ) )

Distinct variable groups:   x, y, A   x, F, y   x, S, y   x, T

Allowed substitution hint:   T( y)

Proof of Theorem onovuni

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onovuni.1	. . . 4 |- ( Lim y -> ( A F y ) = U_ x e. y ( A F x ) )
2		vex 3203	. . . . 5 |- y e. _V
3		oveq2 6658	. . . . . 6 |- ( z = y -> ( A F z ) = ( A F y ) )
4		eqid 2622	. . . . . 6 |- ( z e. _V |-> ( A F z ) ) = ( z e. _V |-> ( A F z ) )
5		ovex 6678	. . . . . 6 |- ( A F y ) e. _V
6	3, 4, 5	fvmpt 6282	. . . . 5 |- ( y e. _V -> ( ( z e. _V |-> ( A F z ) ) ` y ) = ( A F y ) )
7	2, 6	ax-mp 5	. . . 4 |- ( ( z e. _V |-> ( A F z ) ) ` y ) = ( A F y )
8		vex 3203	. . . . . . 7 |- x e. _V
9		oveq2 6658	. . . . . . . 8 |- ( z = x -> ( A F z ) = ( A F x ) )
10		ovex 6678	. . . . . . . 8 |- ( A F x ) e. _V
11	9, 4, 10	fvmpt 6282	. . . . . . 7 |- ( x e. _V -> ( ( z e. _V |-> ( A F z ) ) ` x ) = ( A F x ) )
12	8, 11	ax-mp 5	. . . . . 6 |- ( ( z e. _V |-> ( A F z ) ) ` x ) = ( A F x )
13	12	a1i 11	. . . . 5 |- ( x e. y -> ( ( z e. _V |-> ( A F z ) ) ` x ) = ( A F x ) )
14	13	iuneq2i 4539	. . . 4 |- U_ x e. y ( ( z e. _V |-> ( A F z ) ) ` x ) = U_ x e. y ( A F x )
15	1, 7, 14	3eqtr4g 2681	. . 3 |- ( Lim y -> ( ( z e. _V |-> ( A F z ) ) ` y ) = U_ x e. y ( ( z e. _V |-> ( A F z ) ) ` x ) )
16		onovuni.2	. . . 4 |- ( ( x e. On /\ y e. On /\ x C_ y ) -> ( A F x ) C_ ( A F y ) )
17	16, 12, 7	3sstr4g 3646	. . 3 |- ( ( x e. On /\ y e. On /\ x C_ y ) -> ( ( z e. _V |-> ( A F z ) ) ` x ) C_ ( ( z e. _V |-> ( A F z ) ) ` y ) )
18	15, 17	onfununi 7438	. 2 |- ( ( S e. T /\ S C_ On /\ S =/= (/) ) -> ( ( z e. _V |-> ( A F z ) ) ` U. S ) = U_ x e. S ( ( z e. _V |-> ( A F z ) ) ` x ) )
19		uniexg 6955	. . . 4 |- ( S e. T -> U. S e. _V )
20		oveq2 6658	. . . . 5 |- ( z = U. S -> ( A F z ) = ( A F U. S ) )
21		ovex 6678	. . . . 5 |- ( A F U. S ) e. _V
22	20, 4, 21	fvmpt 6282	. . . 4 |- ( U. S e. _V -> ( ( z e. _V |-> ( A F z ) ) ` U. S ) = ( A F U. S ) )
23	19, 22	syl 17	. . 3 |- ( S e. T -> ( ( z e. _V |-> ( A F z ) ) ` U. S ) = ( A F U. S ) )
24	23	3ad2ant1 1082	. 2 |- ( ( S e. T /\ S C_ On /\ S =/= (/) ) -> ( ( z e. _V |-> ( A F z ) ) ` U. S ) = ( A F U. S ) )
25	12	a1i 11	. . . 4 |- ( x e. S -> ( ( z e. _V |-> ( A F z ) ) ` x ) = ( A F x ) )
26	25	iuneq2i 4539	. . 3 |- U_ x e. S ( ( z e. _V |-> ( A F z ) ) ` x ) = U_ x e. S ( A F x )
27	26	a1i 11	. 2 |- ( ( S e. T /\ S C_ On /\ S =/= (/) ) -> U_ x e. S ( ( z e. _V |-> ( A F z ) ) ` x ) = U_ x e. S ( A F x ) )
28	18, 24, 27	3eqtr3d 2664	1 |- ( ( S e. T /\ S C_ On /\ S =/= (/) ) -> ( A F U. S ) = U_ x e. S ( A F x ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   Oncon0 5723   Lim wlim 5724   ` cfv 5888  (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-sep 4781  ax-nul 4789  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-rab 2921  df-v 3202  df-sbc 3436  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-ord 5726  df-on 5727  df-lim 5728  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  onoviun  7440