Theorem mrcun 16282

Description: Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	|- F = (mrCls ` C )

Assertion

Ref	Expression
mrcun	|- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F ` ( U u. V ) ) = ( F ` ( ( F ` U ) u. ( F ` V ) ) ) )

Proof of Theorem mrcun

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> C e. (Moore ` X ) )
2		mre1cl 16254	. . . . . . 7 |- ( C e. (Moore ` X ) -> X e. C )
3		elpw2g 4827	. . . . . . 7 |- ( X e. C -> ( U e. ~P X <-> U C_ X ) )
4	2, 3	syl 17	. . . . . 6 |- ( C e. (Moore ` X ) -> ( U e. ~P X <-> U C_ X ) )
5	4	biimpar 502	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> U e. ~P X )
6	5	3adant3 1081	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> U e. ~P X )
7		elpw2g 4827	. . . . . . 7 |- ( X e. C -> ( V e. ~P X <-> V C_ X ) )
8	2, 7	syl 17	. . . . . 6 |- ( C e. (Moore ` X ) -> ( V e. ~P X <-> V C_ X ) )
9	8	biimpar 502	. . . . 5 |- ( ( C e. (Moore ` X ) /\ V C_ X ) -> V e. ~P X )
10	9	3adant2 1080	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> V e. ~P X )
11		prssi 4353	. . . 4 |- ( ( U e. ~P X /\ V e. ~P X ) -> { U , V } C_ ~P X )
12	6, 10, 11	syl2anc 693	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> { U , V } C_ ~P X )
13		mrcfval.f	. . . 4 |- F = (mrCls ` C )
14	13	mrcuni 16281	. . 3 |- ( ( C e. (Moore ` X ) /\ { U , V } C_ ~P X ) -> ( F ` U. { U , V } ) = ( F ` U. ( F " { U , V } ) ) )
15	1, 12, 14	syl2anc 693	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F ` U. { U , V } ) = ( F ` U. ( F " { U , V } ) ) )
16		uniprg 4450	. . . 4 |- ( ( U e. ~P X /\ V e. ~P X ) -> U. { U , V } = ( U u. V ) )
17	6, 10, 16	syl2anc 693	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> U. { U , V } = ( U u. V ) )
18	17	fveq2d 6195	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F ` U. { U , V } ) = ( F ` ( U u. V ) ) )
19	13	mrcf 16269	. . . . . . . 8 |- ( C e. (Moore ` X ) -> F : ~P X --> C )
20		ffn 6045	. . . . . . . 8 |- ( F : ~P X --> C -> F Fn ~P X )
21	19, 20	syl 17	. . . . . . 7 |- ( C e. (Moore ` X ) -> F Fn ~P X )
22	21	3ad2ant1 1082	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> F Fn ~P X )
23		fnimapr 6262	. . . . . 6 |- ( ( F Fn ~P X /\ U e. ~P X /\ V e. ~P X ) -> ( F " { U , V } ) = { ( F ` U ) , ( F ` V ) } )
24	22, 6, 10, 23	syl3anc 1326	. . . . 5 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F " { U , V } ) = { ( F ` U ) , ( F ` V ) } )
25	24	unieqd 4446	. . . 4 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> U. ( F " { U , V } ) = U. { ( F ` U ) , ( F ` V ) } )
26		fvex 6201	. . . . 5 |- ( F ` U ) e. _V
27		fvex 6201	. . . . 5 |- ( F ` V ) e. _V
28	26, 27	unipr 4449	. . . 4 |- U. { ( F ` U ) , ( F ` V ) } = ( ( F ` U ) u. ( F ` V ) )
29	25, 28	syl6eq 2672	. . 3 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> U. ( F " { U , V } ) = ( ( F ` U ) u. ( F ` V ) ) )
30	29	fveq2d 6195	. 2 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F ` U. ( F " { U , V } ) ) = ( F ` ( ( F ` U ) u. ( F ` V ) ) ) )
31	15, 18, 30	3eqtr3d 2664	1 |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F ` ( U u. V ) ) = ( F ` ( ( F ` U ) u. ( F ` V ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {cpr 4179   U.cuni 4436   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243

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  ax-un 6949

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-csb 3534  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-int 4476  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by: (None)