Theorem canth4 9469

Description: An "effective" form of Cantor's theorem canth 6608. For any function F from the powerset of A to A, there are two definable sets B and C which witness non-injectivity of F. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)

Hypotheses

Ref	Expression
canth4.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
canth4.2	|- B = U. dom W
canth4.3	|- C = ( `' ( W ` B ) " { ( F ` B ) } )

Assertion

Ref	Expression
canth4	|- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( B C_ A /\ C C. B /\ ( F ` B ) = ( F ` C ) ) )

Distinct variable groups:   x, r, y, A   B, r, x, y   D, r, x, y   F, r, x, y   V, r, x, y   y, C   W, r, x, y

Allowed substitution hints:   C( x, r)

Proof of Theorem canth4

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8 |- B = B
2		eqid 2622	. . . . . . . 8 |- ( W ` B ) = ( W ` B )
3	1, 2	pm3.2i 471	. . . . . . 7 |- ( B = B /\ ( W ` B ) = ( W ` B ) )
4		canth4.1	. . . . . . . 8 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x ( F ` ( `' r " { y } ) ) = y ) ) }
5		elex 3212	. . . . . . . . 9 |- ( A e. V -> A e. _V )
6	5	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> A e. _V )
7		simpl2 1065	. . . . . . . . 9 |- ( ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) /\ x e. ( ~P A i^i dom card ) ) -> F : D --> A )
8		simp3 1063	. . . . . . . . . 10 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( ~P A i^i dom card ) C_ D )
9	8	sselda 3603	. . . . . . . . 9 |- ( ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) /\ x e. ( ~P A i^i dom card ) ) -> x e. D )
10	7, 9	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) /\ x e. ( ~P A i^i dom card ) ) -> ( F ` x ) e. A )
11		canth4.2	. . . . . . . 8 |- B = U. dom W
12	4, 6, 10, 11	fpwwe 9468	. . . . . . 7 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( ( B W ( W ` B ) /\ ( F ` B ) e. B ) <-> ( B = B /\ ( W ` B ) = ( W ` B ) ) ) )
13	3, 12	mpbiri 248	. . . . . 6 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( B W ( W ` B ) /\ ( F ` B ) e. B ) )
14	13	simpld 475	. . . . 5 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> B W ( W ` B ) )
15	4, 6	fpwwelem 9467	. . . . 5 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( B W ( W ` B ) <-> ( ( B C_ A /\ ( W ` B ) C_ ( B X. B ) ) /\ ( ( W ` B ) We B /\ A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y ) ) ) )
16	14, 15	mpbid 222	. . . 4 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( ( B C_ A /\ ( W ` B ) C_ ( B X. B ) ) /\ ( ( W ` B ) We B /\ A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y ) ) )
17	16	simpld 475	. . 3 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( B C_ A /\ ( W ` B ) C_ ( B X. B ) ) )
18	17	simpld 475	. 2 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> B C_ A )
19		canth4.3	. . . . 5 |- C = ( `' ( W ` B ) " { ( F ` B ) } )
20		cnvimass 5485	. . . . 5 |- ( `' ( W ` B ) " { ( F ` B ) } ) C_ dom ( W ` B )
21	19, 20	eqsstri 3635	. . . 4 |- C C_ dom ( W ` B )
22	17	simprd 479	. . . . . 6 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( W ` B ) C_ ( B X. B ) )
23		dmss 5323	. . . . . 6 |- ( ( W ` B ) C_ ( B X. B ) -> dom ( W ` B ) C_ dom ( B X. B ) )
24	22, 23	syl 17	. . . . 5 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> dom ( W ` B ) C_ dom ( B X. B ) )
25		dmxpid 5345	. . . . 5 |- dom ( B X. B ) = B
26	24, 25	syl6sseq 3651	. . . 4 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> dom ( W ` B ) C_ B )
27	21, 26	syl5ss 3614	. . 3 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> C C_ B )
28	13	simprd 479	. . 3 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( F ` B ) e. B )
29	16	simprd 479	. . . . . . 7 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( ( W ` B ) We B /\ A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y ) )
30	29	simpld 475	. . . . . 6 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( W ` B ) We B )
31		weso 5105	. . . . . 6 |- ( ( W ` B ) We B -> ( W ` B ) Or B )
32	30, 31	syl 17	. . . . 5 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( W ` B ) Or B )
33		sonr 5056	. . . . 5 |- ( ( ( W ` B ) Or B /\ ( F ` B ) e. B ) -> -. ( F ` B ) ( W ` B ) ( F ` B ) )
34	32, 28, 33	syl2anc 693	. . . 4 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> -. ( F ` B ) ( W ` B ) ( F ` B ) )
35	19	eleq2i 2693	. . . . 5 |- ( ( F ` B ) e. C <-> ( F ` B ) e. ( `' ( W ` B ) " { ( F ` B ) } ) )
36		fvex 6201	. . . . . 6 |- ( F ` B ) e. _V
37	36	eliniseg 5494	. . . . . 6 |- ( ( F ` B ) e. _V -> ( ( F ` B ) e. ( `' ( W ` B ) " { ( F ` B ) } ) <-> ( F ` B ) ( W ` B ) ( F ` B ) ) )
38	36, 37	ax-mp 5	. . . . 5 |- ( ( F ` B ) e. ( `' ( W ` B ) " { ( F ` B ) } ) <-> ( F ` B ) ( W ` B ) ( F ` B ) )
39	35, 38	bitri 264	. . . 4 |- ( ( F ` B ) e. C <-> ( F ` B ) ( W ` B ) ( F ` B ) )
40	34, 39	sylnibr 319	. . 3 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> -. ( F ` B ) e. C )
41	27, 28, 40	ssnelpssd 3719	. 2 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> C C. B )
42	29	simprd 479	. . . 4 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y )
43		sneq 4187	. . . . . . . . 9 |- ( y = ( F ` B ) -> { y } = { ( F ` B ) } )
44	43	imaeq2d 5466	. . . . . . . 8 |- ( y = ( F ` B ) -> ( `' ( W ` B ) " { y } ) = ( `' ( W ` B ) " { ( F ` B ) } ) )
45	44, 19	syl6eqr 2674	. . . . . . 7 |- ( y = ( F ` B ) -> ( `' ( W ` B ) " { y } ) = C )
46	45	fveq2d 6195	. . . . . 6 |- ( y = ( F ` B ) -> ( F ` ( `' ( W ` B ) " { y } ) ) = ( F ` C ) )
47		id 22	. . . . . 6 |- ( y = ( F ` B ) -> y = ( F ` B ) )
48	46, 47	eqeq12d 2637	. . . . 5 |- ( y = ( F ` B ) -> ( ( F ` ( `' ( W ` B ) " { y } ) ) = y <-> ( F ` C ) = ( F ` B ) ) )
49	48	rspcv 3305	. . . 4 |- ( ( F ` B ) e. B -> ( A. y e. B ( F ` ( `' ( W ` B ) " { y } ) ) = y -> ( F ` C ) = ( F ` B ) ) )
50	28, 42, 49	sylc 65	. . 3 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( F ` C ) = ( F ` B ) )
51	50	eqcomd 2628	. 2 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( F ` B ) = ( F ` C ) )
52	18, 41, 51	3jca 1242	1 |- ( ( A e. V /\ F : D --> A /\ ( ~P A i^i dom card ) C_ D ) -> ( B C_ A /\ C C. B /\ ( F ` B ) = ( F ` C ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   C. wpss 3575   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   {copab 4712   Or wor 5034   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888   cardccrd 8761

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-int 4476  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-1st 7168  df-wrecs 7407  df-recs 7468  df-en 7956  df-oi 8415  df-card 8765

This theorem is referenced by:  canthnumlem  9470  canthp1lem2  9475