Theorem mreunirn 16261

Description: Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
mreunirn	|- ( C e. U. ran Moore <-> C e. (Moore ` U. C ) )

Proof of Theorem mreunirn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmre 16251	. . . 4 |- Moore Fn _V
2		fnunirn 6511	. . . 4 |- (Moore Fn _V -> ( C e. U. ran Moore <-> E. x e. _V C e. (Moore ` x ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( C e. U. ran Moore <-> E. x e. _V C e. (Moore ` x ) )
4		mreuni 16260	. . . . . . 7 |- ( C e. (Moore ` x ) -> U. C = x )
5	4	fveq2d 6195	. . . . . 6 |- ( C e. (Moore ` x ) -> (Moore ` U. C ) = (Moore ` x ) )
6	5	eleq2d 2687	. . . . 5 |- ( C e. (Moore ` x ) -> ( C e. (Moore ` U. C ) <-> C e. (Moore ` x ) ) )
7	6	ibir 257	. . . 4 |- ( C e. (Moore ` x ) -> C e. (Moore ` U. C ) )
8	7	rexlimivw 3029	. . 3 |- ( E. x e. _V C e. (Moore ` x ) -> C e. (Moore ` U. C ) )
9	3, 8	sylbi 207	. 2 |- ( C e. U. ran Moore -> C e. (Moore ` U. C ) )
10		fvssunirn 6217	. . 3 |- (Moore ` U. C ) C_ U. ran Moore
11	10	sseli 3599	. 2 |- ( C e. (Moore ` U. C ) -> C e. U. ran Moore )
12	9, 11	impbii 199	1 |- ( C e. U. ran Moore <-> C e. (Moore ` U. C ) )

Syntax hints:   <-> wb 196   e. wcel 1990   E.wrex 2913   _Vcvv 3200   U.cuni 4436   ran crn 5115   Fn wfn 5883   ` cfv 5888  Moorecmre 16242

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-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-mre 16246

This theorem is referenced by:  fnmrc  16267  mrcfval  16268