Theorem fpwwe2lem11 9462

Description: Lemma for fpwwe2 9465. (Contributed by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
fpwwe2.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
fpwwe2.2	|- ( ph -> A e. _V )
fpwwe2.3	|- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
fpwwe2.4	|- X = U. dom W

Assertion

Ref	Expression
fpwwe2lem11	|- ( ph -> W : dom W --> ~P ( X X. X ) )

Distinct variable groups:   y, u, r, x, F   X, r, u, x, y   ph, r, u, x, y   A, r, x   W, r, u, x, y

Allowed substitution hints:   A( y, u)

Proof of Theorem fpwwe2lem11

Dummy variables s t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . . 6 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
2	1	relopabi 5245	. . . . 5 |- Rel W
3	2	a1i 11	. . . 4 |- ( ph -> Rel W )
4		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ s = ( t i^i ( w X. w ) ) ) ) -> s = ( t i^i ( w X. w ) ) )
5		fpwwe2.2	. . . . . . . . . . . . . . 15 |- ( ph -> A e. _V )
6	1, 5	fpwwe2lem2 9454	. . . . . . . . . . . . . 14 |- ( ph -> ( w W t <-> ( ( w C_ A /\ t C_ ( w X. w ) ) /\ ( t We w /\ A. y e. w [. ( `' t " { y } ) / u ]. ( u F ( t i^i ( u X. u ) ) ) = y ) ) ) )
7	6	simprbda 653	. . . . . . . . . . . . 13 |- ( ( ph /\ w W t ) -> ( w C_ A /\ t C_ ( w X. w ) ) )
8	7	simprd 479	. . . . . . . . . . . 12 |- ( ( ph /\ w W t ) -> t C_ ( w X. w ) )
9	8	adantrl 752	. . . . . . . . . . 11 |- ( ( ph /\ ( w W s /\ w W t ) ) -> t C_ ( w X. w ) )
10	9	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ s = ( t i^i ( w X. w ) ) ) ) -> t C_ ( w X. w ) )
11		df-ss 3588	. . . . . . . . . 10 |- ( t C_ ( w X. w ) <-> ( t i^i ( w X. w ) ) = t )
12	10, 11	sylib 208	. . . . . . . . 9 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ s = ( t i^i ( w X. w ) ) ) ) -> ( t i^i ( w X. w ) ) = t )
13	4, 12	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ s = ( t i^i ( w X. w ) ) ) ) -> s = t )
14		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ t = ( s i^i ( w X. w ) ) ) ) -> t = ( s i^i ( w X. w ) ) )
15	1, 5	fpwwe2lem2 9454	. . . . . . . . . . . . . 14 |- ( ph -> ( w W s <-> ( ( w C_ A /\ s C_ ( w X. w ) ) /\ ( s We w /\ A. y e. w [. ( `' s " { y } ) / u ]. ( u F ( s i^i ( u X. u ) ) ) = y ) ) ) )
16	15	simprbda 653	. . . . . . . . . . . . 13 |- ( ( ph /\ w W s ) -> ( w C_ A /\ s C_ ( w X. w ) ) )
17	16	simprd 479	. . . . . . . . . . . 12 |- ( ( ph /\ w W s ) -> s C_ ( w X. w ) )
18	17	adantrr 753	. . . . . . . . . . 11 |- ( ( ph /\ ( w W s /\ w W t ) ) -> s C_ ( w X. w ) )
19	18	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ t = ( s i^i ( w X. w ) ) ) ) -> s C_ ( w X. w ) )
20		df-ss 3588	. . . . . . . . . 10 |- ( s C_ ( w X. w ) <-> ( s i^i ( w X. w ) ) = s )
21	19, 20	sylib 208	. . . . . . . . 9 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ t = ( s i^i ( w X. w ) ) ) ) -> ( s i^i ( w X. w ) ) = s )
22	14, 21	eqtr2d 2657	. . . . . . . 8 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( w C_ w /\ t = ( s i^i ( w X. w ) ) ) ) -> s = t )
23	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( w W s /\ w W t ) ) -> A e. _V )
24		fpwwe2.3	. . . . . . . . . 10 |- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
25	24	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ ( w W s /\ w W t ) ) /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
26		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( w W s /\ w W t ) ) -> w W s )
27		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( w W s /\ w W t ) ) -> w W t )
28	1, 23, 25, 26, 27	fpwwe2lem10 9461	. . . . . . . 8 |- ( ( ph /\ ( w W s /\ w W t ) ) -> ( ( w C_ w /\ s = ( t i^i ( w X. w ) ) ) \/ ( w C_ w /\ t = ( s i^i ( w X. w ) ) ) ) )
29	13, 22, 28	mpjaodan 827	. . . . . . 7 |- ( ( ph /\ ( w W s /\ w W t ) ) -> s = t )
30	29	ex 450	. . . . . 6 |- ( ph -> ( ( w W s /\ w W t ) -> s = t ) )
31	30	alrimiv 1855	. . . . 5 |- ( ph -> A. t ( ( w W s /\ w W t ) -> s = t ) )
32	31	alrimivv 1856	. . . 4 |- ( ph -> A. w A. s A. t ( ( w W s /\ w W t ) -> s = t ) )
33		dffun2 5898	. . . 4 |- ( Fun W <-> ( Rel W /\ A. w A. s A. t ( ( w W s /\ w W t ) -> s = t ) ) )
34	3, 32, 33	sylanbrc 698	. . 3 |- ( ph -> Fun W )
35		funfn 5918	. . 3 |- ( Fun W <-> W Fn dom W )
36	34, 35	sylib 208	. 2 |- ( ph -> W Fn dom W )
37		vex 3203	. . . . 5 |- s e. _V
38	37	elrn 5366	. . . 4 |- ( s e. ran W <-> E. w w W s )
39	2	releldmi 5362	. . . . . . . . . . . 12 |- ( w W s -> w e. dom W )
40	39	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ w W s ) -> w e. dom W )
41		elssuni 4467	. . . . . . . . . . 11 |- ( w e. dom W -> w C_ U. dom W )
42	40, 41	syl 17	. . . . . . . . . 10 |- ( ( ph /\ w W s ) -> w C_ U. dom W )
43		fpwwe2.4	. . . . . . . . . 10 |- X = U. dom W
44	42, 43	syl6sseqr 3652	. . . . . . . . 9 |- ( ( ph /\ w W s ) -> w C_ X )
45		xpss12 5225	. . . . . . . . 9 |- ( ( w C_ X /\ w C_ X ) -> ( w X. w ) C_ ( X X. X ) )
46	44, 44, 45	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ w W s ) -> ( w X. w ) C_ ( X X. X ) )
47	17, 46	sstrd 3613	. . . . . . 7 |- ( ( ph /\ w W s ) -> s C_ ( X X. X ) )
48	47	ex 450	. . . . . 6 |- ( ph -> ( w W s -> s C_ ( X X. X ) ) )
49		selpw 4165	. . . . . 6 |- ( s e. ~P ( X X. X ) <-> s C_ ( X X. X ) )
50	48, 49	syl6ibr 242	. . . . 5 |- ( ph -> ( w W s -> s e. ~P ( X X. X ) ) )
51	50	exlimdv 1861	. . . 4 |- ( ph -> ( E. w w W s -> s e. ~P ( X X. X ) ) )
52	38, 51	syl5bi 232	. . 3 |- ( ph -> ( s e. ran W -> s e. ~P ( X X. X ) ) )
53	52	ssrdv 3609	. 2 |- ( ph -> ran W C_ ~P ( X X. X ) )
54		df-f 5892	. 2 |- ( W : dom W --> ~P ( X X. X ) <-> ( W Fn dom W /\ ran W C_ ~P ( X X. X ) ) )
55	36, 53, 54	sylanbrc 698	1 |- ( ph -> W : dom W --> ~P ( X X. X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->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-rmo 2920  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-wrecs 7407  df-recs 7468  df-oi 8415

This theorem is referenced by:  fpwwe2lem13  9464  fpwwe2  9465