Theorem fpwwe 9089

Description: Given any function F from the powerset of A to A, canth2 7743 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset <. X , ( W ` X ) >. which "agrees" with F in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 8479. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Hypotheses

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

Assertion

Ref	Expression
fpwwe	|- ( ph -> ( ( Y W R /\ ( F ` Y ) e. Y ) <-> ( Y = X /\ R = ( W ` X ) ) ) )

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

Allowed substitution hint:   A( y)

Proof of Theorem fpwwe

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6311	. . . . . 6 |- ( Y ( F o. 1st ) R ) = ( ( F o. 1st ) ` <. Y , R >. )
2		fo1st 6832	. . . . . . . 8 |- 1st : _V -onto-> _V
3		fofn 5808	. . . . . . . 8 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . . . . . 7 |- 1st Fn _V
5		opex 4664	. . . . . . 7 |- <. Y , R >. e. _V
6		fvco2 5955	. . . . . . 7 |- ( ( 1st Fn _V /\ <. Y , R >. e. _V ) -> ( ( F o. 1st ) ` <. Y , R >. ) = ( F ` ( 1st ` <. Y , R >. ) ) )
7	4, 5, 6	mp2an 686	. . . . . 6 |- ( ( F o. 1st ) ` <. Y , R >. ) = ( F ` ( 1st ` <. Y , R >. ) )
8	1, 7	eqtri 2493	. . . . 5 |- ( Y ( F o. 1st ) R ) = ( F ` ( 1st ` <. Y , R >. ) )
9		fpwwe.1	. . . . . . . 8 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
10	9	bropaex12 4913	. . . . . . 7 |- ( Y W R -> ( Y e. _V /\ R e. _V ) )
11		op1stg 6824	. . . . . . 7 |- ( ( Y e. _V /\ R e. _V ) -> ( 1st ` <. Y , R >. ) = Y )
12	10, 11	syl 17	. . . . . 6 |- ( Y W R -> ( 1st ` <. Y , R >. ) = Y )
13	12	fveq2d 5883	. . . . 5 |- ( Y W R -> ( F ` ( 1st ` <. Y , R >. ) ) = ( F ` Y ) )
14	8, 13	syl5eq 2517	. . . 4 |- ( Y W R -> ( Y ( F o. 1st ) R ) = ( F ` Y ) )
15	14	eleq1d 2533	. . 3 |- ( Y W R -> ( ( Y ( F o. 1st ) R ) e. Y <-> ( F ` Y ) e. Y ) )
16	15	pm5.32i 649	. 2 |- ( ( Y W R /\ ( Y ( F o. 1st ) R ) e. Y ) <-> ( Y W R /\ ( F ` Y ) e. Y ) )
17		vex 3034	. . . . . . . . . 10 |- r e. _V
18		cnvexg 6758	. . . . . . . . . 10 |- ( r e. _V -> `' r e. _V )
19		imaexg 6749	. . . . . . . . . 10 |- ( `' r e. _V -> ( `' r " { y } ) e. _V )
20	17, 18, 19	mp2b 10	. . . . . . . . 9 |- ( `' r " { y } ) e. _V
21		vex 3034	. . . . . . . . . . . 12 |- u e. _V
22	17	inex1 4537	. . . . . . . . . . . 12 |- ( r i^i ( u X. u ) ) e. _V
23	21, 22	algrflem 6924	. . . . . . . . . . 11 |- ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = ( F ` u )
24		fveq2 5879	. . . . . . . . . . 11 |- ( u = ( `' r " { y } ) -> ( F ` u ) = ( F ` ( `' r " { y } ) ) )
25	23, 24	syl5eq 2517	. . . . . . . . . 10 |- ( u = ( `' r " { y } ) -> ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = ( F ` ( `' r " { y } ) ) )
26	25	eqeq1d 2473	. . . . . . . . 9 |- ( u = ( `' r " { y } ) -> ( ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y <-> ( F ` ( `' r " { y } ) ) = y ) )
27	20, 26	sbcie 3290	. . . . . . . 8 |- ( [. ( `' r " { y } ) / u ]. ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y <-> ( F ` ( `' r " { y } ) ) = y )
28	27	ralbii 2823	. . . . . . 7 |- ( A. y e. x [. ( `' r " { y } ) / u ]. ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y <-> A. y e. x ( F ` ( `' r " { y } ) ) = y )
29	28	anbi2i 708	. . . . . 6 |- ( ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y ) <-> ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) )
30	29	anbi2i 708	. . . . 5 |- ( ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y ) ) <-> ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) )
31	30	opabbii 4460	. . . 4 |- { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y ) ) } = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
32	9, 31	eqtr4i 2496	. . 3 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u ( F o. 1st ) ( r i^i ( u X. u ) ) ) = y ) ) }
33		fpwwe.2	. . 3 |- ( ph -> A e. _V )
34		vex 3034	. . . . 5 |- x e. _V
35	34, 17	algrflem 6924	. . . 4 |- ( x ( F o. 1st ) r ) = ( F ` x )
36		simp1 1030	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x C_ A )
37		selpw 3949	. . . . . . 7 |- ( x e. ~P A <-> x C_ A )
38	36, 37	sylibr 217	. . . . . 6 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x e. ~P A )
39		19.8a 1955	. . . . . . . 8 |- ( r We x -> E. r r We x )
40	39	3ad2ant3 1053	. . . . . . 7 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> E. r r We x )
41		ween 8484	. . . . . . 7 |- ( x e. dom card <-> E. r r We x )
42	40, 41	sylibr 217	. . . . . 6 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x e. dom card )
43	38, 42	elind 3609	. . . . 5 |- ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) -> x e. ( ~P A i^i dom card ) )
44		fpwwe.3	. . . . 5 |- ( ( ph /\ x e. ( ~P A i^i dom card ) ) -> ( F ` x ) e. A )
45	43, 44	sylan2 482	. . . 4 |- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( F ` x ) e. A )
46	35, 45	syl5eqel 2553	. . 3 |- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x ( F o. 1st ) r ) e. A )
47		fpwwe.4	. . 3 |- X = U. dom W
48	32, 33, 46, 47	fpwwe2 9086	. 2 |- ( ph -> ( ( Y W R /\ ( Y ( F o. 1st ) R ) e. Y ) <-> ( Y = X /\ R = ( W ` X ) ) ) )
49	16, 48	syl5bbr 267	1 |- ( ph -> ( ( Y W R /\ ( F ` Y ) e. Y ) <-> ( Y = X /\ R = ( W ` X ) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   A.wral 2756   _Vcvv 3031   [.wsbc 3255   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395   {copab 4453   We wwe 4797   X. cxp 4837   `'ccnv 4838   dom cdm 4839   "cima 4842   o. ccom 4843   Fn wfn 5584   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   1stc1st 6810   cardccrd 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-1st 6812  df-wrecs 7046  df-recs 7108  df-en 7588  df-oi 8043  df-card 8391

This theorem is referenced by:  canth4  9090  canthnumlem  9091  canthp1lem2  9096