Theorem pwfseqlem4 9484

Description: Lemma for pwfseq 9486. Derive a final contradiction from the function F in pwfseqlem3 9482. Applying fpwwe2 9465 to it, we get a certain maximal well-ordered subset Z, but the defining property ( Z F ( W ` Z ) ) e. Z contradicts our assumption on F, so we are reduced to the case of Z finite. This too is a contradiction, though, because Z and its preimage under ( W ` Z ) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses

Ref	Expression
pwfseqlem4.g	|- ( ph -> G : ~P A -1-1-> U_ n e. om ( A ^m n ) )
pwfseqlem4.x	|- ( ph -> X C_ A )
pwfseqlem4.h	|- ( ph -> H : om -1-1-onto-> X )
pwfseqlem4.ps	|- ( ps <-> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) /\ om ~<_ x ) )
pwfseqlem4.k	|- ( ( ph /\ ps ) -> K : U_ n e. om ( x ^m n ) -1-1-> x )
pwfseqlem4.d	|- D = ( G ` { w e. x | ( ( `' K ` w ) e. ran G /\ -. w e. ( `' G ` ( `' K ` w ) ) ) } )
pwfseqlem4.f	|- F = ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
pwfseqlem4.w	|- W = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. b e. a [. ( `' s " { b } ) / v ]. ( v F ( s i^i ( v X. v ) ) ) = b ) ) }
pwfseqlem4.z	|- Z = U. dom W

Assertion

Ref	Expression
pwfseqlem4	|- -. ph

Distinct variable groups:   n, r, w, x, z   D, n, z   a, b, s, v, F   w, G   w, K   r, a, x, z, H, b, s, v   n, a, ph, b, s, v, r, x, z   ps, n, z   A, a, n, r, s, x, z   W, a, b, s, v   Z, a, b, s, v

Allowed substitution hints:   ph( w)   ps( x, w, v, s, r, a, b)   A( w, v, b)   D( x, w, v, s, r, a, b)   F( x, z, w, n, r)   G( x, z, v, n, s, r, a, b)   H( w, n)   K( x, z, v, n, s, r, a, b)   W( x, z, w, n, r)   X( x, z, w, v, n, s, r, a, b)   Z( x, z, w, n, r)

Proof of Theorem pwfseqlem4

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . . . 11 |- Z = Z
2		eqid 2622	. . . . . . . . . . 11 |- ( W ` Z ) = ( W ` Z )
3	1, 2	pm3.2i 471	. . . . . . . . . 10 |- ( Z = Z /\ ( W ` Z ) = ( W ` Z ) )
4		pwfseqlem4.w	. . . . . . . . . . 11 |- W = { <. a , s >. | ( ( a C_ A /\ s C_ ( a X. a ) ) /\ ( s We a /\ A. b e. a [. ( `' s " { b } ) / v ]. ( v F ( s i^i ( v X. v ) ) ) = b ) ) }
5		pwfseqlem4.g	. . . . . . . . . . . . 13 |- ( ph -> G : ~P A -1-1-> U_ n e. om ( A ^m n ) )
6		omex 8540	. . . . . . . . . . . . . 14 |- om e. _V
7		ovex 6678	. . . . . . . . . . . . . 14 |- ( A ^m n ) e. _V
8	6, 7	iunex 7147	. . . . . . . . . . . . 13 |- U_ n e. om ( A ^m n ) e. _V
9		f1dmex 7136	. . . . . . . . . . . . 13 |- ( ( G : ~P A -1-1-> U_ n e. om ( A ^m n ) /\ U_ n e. om ( A ^m n ) e. _V ) -> ~P A e. _V )
10	5, 8, 9	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> ~P A e. _V )
11		pwexb 6975	. . . . . . . . . . . 12 |- ( A e. _V <-> ~P A e. _V )
12	10, 11	sylibr 224	. . . . . . . . . . 11 |- ( ph -> A e. _V )
13		pwfseqlem4.x	. . . . . . . . . . . 12 |- ( ph -> X C_ A )
14		pwfseqlem4.h	. . . . . . . . . . . 12 |- ( ph -> H : om -1-1-onto-> X )
15		pwfseqlem4.ps	. . . . . . . . . . . 12 |- ( ps <-> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) /\ om ~<_ x ) )
16		pwfseqlem4.k	. . . . . . . . . . . 12 |- ( ( ph /\ ps ) -> K : U_ n e. om ( x ^m n ) -1-1-> x )
17		pwfseqlem4.d	. . . . . . . . . . . 12 |- D = ( G ` { w e. x | ( ( `' K ` w ) e. ran G /\ -. w e. ( `' G ` ( `' K ` w ) ) ) } )
18		pwfseqlem4.f	. . . . . . . . . . . 12 |- F = ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
19	5, 13, 14, 15, 16, 17, 18	pwfseqlem4a 9483	. . . . . . . . . . 11 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( a F s ) e. A )
20		pwfseqlem4.z	. . . . . . . . . . 11 |- Z = U. dom W
21	4, 12, 19, 20	fpwwe2 9465	. . . . . . . . . 10 |- ( ph -> ( ( Z W ( W ` Z ) /\ ( Z F ( W ` Z ) ) e. Z ) <-> ( Z = Z /\ ( W ` Z ) = ( W ` Z ) ) ) )
22	3, 21	mpbiri 248	. . . . . . . . 9 |- ( ph -> ( Z W ( W ` Z ) /\ ( Z F ( W ` Z ) ) e. Z ) )
23	22	simprd 479	. . . . . . . 8 |- ( ph -> ( Z F ( W ` Z ) ) e. Z )
24	22	simpld 475	. . . . . . . . . . . . 13 |- ( ph -> Z W ( W ` Z ) )
25	4, 12	fpwwe2lem2 9454	. . . . . . . . . . . . 13 |- ( ph -> ( Z W ( W ` Z ) <-> ( ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) ) /\ ( ( W ` Z ) We Z /\ A. b e. Z [. ( `' ( W ` Z ) " { b } ) / v ]. ( v F ( ( W ` Z ) i^i ( v X. v ) ) ) = b ) ) ) )
26	24, 25	mpbid 222	. . . . . . . . . . . 12 |- ( ph -> ( ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) ) /\ ( ( W ` Z ) We Z /\ A. b e. Z [. ( `' ( W ` Z ) " { b } ) / v ]. ( v F ( ( W ` Z ) i^i ( v X. v ) ) ) = b ) ) )
27	26	simpld 475	. . . . . . . . . . 11 |- ( ph -> ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) ) )
28	27	simpld 475	. . . . . . . . . 10 |- ( ph -> Z C_ A )
29	12, 28	ssexd 4805	. . . . . . . . 9 |- ( ph -> Z e. _V )
30		sseq1 3626	. . . . . . . . . . . . . 14 |- ( a = Z -> ( a C_ A <-> Z C_ A ) )
31		id 22	. . . . . . . . . . . . . . . 16 |- ( a = Z -> a = Z )
32	31	sqxpeqd 5141	. . . . . . . . . . . . . . 15 |- ( a = Z -> ( a X. a ) = ( Z X. Z ) )
33	32	sseq2d 3633	. . . . . . . . . . . . . 14 |- ( a = Z -> ( ( W ` Z ) C_ ( a X. a ) <-> ( W ` Z ) C_ ( Z X. Z ) ) )
34		weeq2 5103	. . . . . . . . . . . . . 14 |- ( a = Z -> ( ( W ` Z ) We a <-> ( W ` Z ) We Z ) )
35	30, 33, 34	3anbi123d 1399	. . . . . . . . . . . . 13 |- ( a = Z -> ( ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) <-> ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) ) )
36	35	anbi2d 740	. . . . . . . . . . . 12 |- ( a = Z -> ( ( ph /\ ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) <-> ( ph /\ ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) ) ) )
37		id 22	. . . . . . . . . . . . . . . 16 |- ( ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) -> ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) )
38	37	3expa 1265	. . . . . . . . . . . . . . 15 |- ( ( ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) ) /\ ( W ` Z ) We Z ) -> ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) )
39	38	adantrr 753	. . . . . . . . . . . . . 14 |- ( ( ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) ) /\ ( ( W ` Z ) We Z /\ A. b e. Z [. ( `' ( W ` Z ) " { b } ) / v ]. ( v F ( ( W ` Z ) i^i ( v X. v ) ) ) = b ) ) -> ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) )
40	26, 39	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) )
41	40	pm4.71i 664	. . . . . . . . . . . 12 |- ( ph <-> ( ph /\ ( Z C_ A /\ ( W ` Z ) C_ ( Z X. Z ) /\ ( W ` Z ) We Z ) ) )
42	36, 41	syl6bbr 278	. . . . . . . . . . 11 |- ( a = Z -> ( ( ph /\ ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) <-> ph ) )
43		oveq1 6657	. . . . . . . . . . . . 13 |- ( a = Z -> ( a F ( W ` Z ) ) = ( Z F ( W ` Z ) ) )
44	43, 31	eleq12d 2695	. . . . . . . . . . . 12 |- ( a = Z -> ( ( a F ( W ` Z ) ) e. a <-> ( Z F ( W ` Z ) ) e. Z ) )
45		breq1 4656	. . . . . . . . . . . 12 |- ( a = Z -> ( a ~< om <-> Z ~< om ) )
46	44, 45	imbi12d 334	. . . . . . . . . . 11 |- ( a = Z -> ( ( ( a F ( W ` Z ) ) e. a -> a ~< om ) <-> ( ( Z F ( W ` Z ) ) e. Z -> Z ~< om ) ) )
47	42, 46	imbi12d 334	. . . . . . . . . 10 |- ( a = Z -> ( ( ( ph /\ ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) -> ( ( a F ( W ` Z ) ) e. a -> a ~< om ) ) <-> ( ph -> ( ( Z F ( W ` Z ) ) e. Z -> Z ~< om ) ) ) )
48		fvex 6201	. . . . . . . . . . 11 |- ( W ` Z ) e. _V
49		sseq1 3626	. . . . . . . . . . . . . 14 |- ( s = ( W ` Z ) -> ( s C_ ( a X. a ) <-> ( W ` Z ) C_ ( a X. a ) ) )
50		weeq1 5102	. . . . . . . . . . . . . 14 |- ( s = ( W ` Z ) -> ( s We a <-> ( W ` Z ) We a ) )
51	49, 50	3anbi23d 1402	. . . . . . . . . . . . 13 |- ( s = ( W ` Z ) -> ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) <-> ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) )
52	51	anbi2d 740	. . . . . . . . . . . 12 |- ( s = ( W ` Z ) -> ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) <-> ( ph /\ ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) ) )
53		oveq2 6658	. . . . . . . . . . . . . 14 |- ( s = ( W ` Z ) -> ( a F s ) = ( a F ( W ` Z ) ) )
54	53	eleq1d 2686	. . . . . . . . . . . . 13 |- ( s = ( W ` Z ) -> ( ( a F s ) e. a <-> ( a F ( W ` Z ) ) e. a ) )
55	54	imbi1d 331	. . . . . . . . . . . 12 |- ( s = ( W ` Z ) -> ( ( ( a F s ) e. a -> a ~< om ) <-> ( ( a F ( W ` Z ) ) e. a -> a ~< om ) ) )
56	52, 55	imbi12d 334	. . . . . . . . . . 11 |- ( s = ( W ` Z ) -> ( ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( ( a F s ) e. a -> a ~< om ) ) <-> ( ( ph /\ ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) -> ( ( a F ( W ` Z ) ) e. a -> a ~< om ) ) ) )
57		omelon 8543	. . . . . . . . . . . . . . 15 |- om e. On
58		onenon 8775	. . . . . . . . . . . . . . 15 |- ( om e. On -> om e. dom card )
59	57, 58	ax-mp 5	. . . . . . . . . . . . . 14 |- om e. dom card
60		simpr3 1069	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> s We a )
61		19.8a 2052	. . . . . . . . . . . . . . . 16 |- ( s We a -> E. s s We a )
62	60, 61	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> E. s s We a )
63		ween 8858	. . . . . . . . . . . . . . 15 |- ( a e. dom card <-> E. s s We a )
64	62, 63	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> a e. dom card )
65		domtri2 8815	. . . . . . . . . . . . . 14 |- ( ( om e. dom card /\ a e. dom card ) -> ( om ~<_ a <-> -. a ~< om ) )
66	59, 64, 65	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( om ~<_ a <-> -. a ~< om ) )
67		nfv 1843	. . . . . . . . . . . . . . . . 17 |- F/ r ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) )
68		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 |- F/_ r a
69		nfmpt22 6723	. . . . . . . . . . . . . . . . . . . 20 |- F/_ r ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
70	18, 69	nfcxfr 2762	. . . . . . . . . . . . . . . . . . 19 |- F/_ r F
71		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 |- F/_ r s
72	68, 70, 71	nfov 6676	. . . . . . . . . . . . . . . . . 18 |- F/_ r ( a F s )
73	72	nfel1 2779	. . . . . . . . . . . . . . . . 17 |- F/ r ( a F s ) e. ( A \ a )
74	67, 73	nfim 1825	. . . . . . . . . . . . . . . 16 |- F/ r ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. ( A \ a ) )
75		sseq1 3626	. . . . . . . . . . . . . . . . . . . 20 |- ( r = s -> ( r C_ ( a X. a ) <-> s C_ ( a X. a ) ) )
76		weeq1 5102	. . . . . . . . . . . . . . . . . . . 20 |- ( r = s -> ( r We a <-> s We a ) )
77	75, 76	3anbi23d 1402	. . . . . . . . . . . . . . . . . . 19 |- ( r = s -> ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) <-> ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) )
78	77	anbi1d 741	. . . . . . . . . . . . . . . . . 18 |- ( r = s -> ( ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) <-> ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) )
79	78	anbi2d 740	. . . . . . . . . . . . . . . . 17 |- ( r = s -> ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) <-> ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) ) )
80		oveq2 6658	. . . . . . . . . . . . . . . . . 18 |- ( r = s -> ( a F r ) = ( a F s ) )
81	80	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( r = s -> ( ( a F r ) e. ( A \ a ) <-> ( a F s ) e. ( A \ a ) ) )
82	79, 81	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( r = s -> ( ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) ) <-> ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. ( A \ a ) ) ) )
83		nfv 1843	. . . . . . . . . . . . . . . . . 18 |- F/ x ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) )
84		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 |- F/_ x a
85		nfmpt21 6722	. . . . . . . . . . . . . . . . . . . . 21 |- F/_ x ( x e. _V , r e. _V |-> if ( x e. Fin , ( H ` ( card ` x ) ) , ( D ` |^| { z e. om | -. ( D ` z ) e. x } ) ) )
86	18, 85	nfcxfr 2762	. . . . . . . . . . . . . . . . . . . 20 |- F/_ x F
87		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 |- F/_ x r
88	84, 86, 87	nfov 6676	. . . . . . . . . . . . . . . . . . 19 |- F/_ x ( a F r )
89	88	nfel1 2779	. . . . . . . . . . . . . . . . . 18 |- F/ x ( a F r ) e. ( A \ a )
90	83, 89	nfim 1825	. . . . . . . . . . . . . . . . 17 |- F/ x ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) )
91		sseq1 3626	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = a -> ( x C_ A <-> a C_ A ) )
92		xpeq12 5134	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x = a /\ x = a ) -> ( x X. x ) = ( a X. a ) )
93	92	anidms 677	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( x = a -> ( x X. x ) = ( a X. a ) )
94	93	sseq2d 3633	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = a -> ( r C_ ( x X. x ) <-> r C_ ( a X. a ) ) )
95		weeq2 5103	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x = a -> ( r We x <-> r We a ) )
96	91, 94, 95	3anbi123d 1399	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = a -> ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) <-> ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) ) )
97		breq2 4657	. . . . . . . . . . . . . . . . . . . . 21 |- ( x = a -> ( om ~<_ x <-> om ~<_ a ) )
98	96, 97	anbi12d 747	. . . . . . . . . . . . . . . . . . . 20 |- ( x = a -> ( ( ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) /\ om ~<_ x ) <-> ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) )
99	15, 98	syl5bb 272	. . . . . . . . . . . . . . . . . . 19 |- ( x = a -> ( ps <-> ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) )
100	99	anbi2d 740	. . . . . . . . . . . . . . . . . 18 |- ( x = a -> ( ( ph /\ ps ) <-> ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) ) )
101		oveq1 6657	. . . . . . . . . . . . . . . . . . 19 |- ( x = a -> ( x F r ) = ( a F r ) )
102		difeq2 3722	. . . . . . . . . . . . . . . . . . 19 |- ( x = a -> ( A \ x ) = ( A \ a ) )
103	101, 102	eleq12d 2695	. . . . . . . . . . . . . . . . . 18 |- ( x = a -> ( ( x F r ) e. ( A \ x ) <-> ( a F r ) e. ( A \ a ) ) )
104	100, 103	imbi12d 334	. . . . . . . . . . . . . . . . 17 |- ( x = a -> ( ( ( ph /\ ps ) -> ( x F r ) e. ( A \ x ) ) <-> ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) ) ) )
105	5, 13, 14, 15, 16, 17, 18	pwfseqlem3 9482	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ps ) -> ( x F r ) e. ( A \ x ) )
106	90, 104, 105	chvar 2262	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( ( a C_ A /\ r C_ ( a X. a ) /\ r We a ) /\ om ~<_ a ) ) -> ( a F r ) e. ( A \ a ) )
107	74, 82, 106	chvar 2262	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> ( a F s ) e. ( A \ a ) )
108	107	eldifbd 3587	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) /\ om ~<_ a ) ) -> -. ( a F s ) e. a )
109	108	expr 643	. . . . . . . . . . . . 13 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( om ~<_ a -> -. ( a F s ) e. a ) )
110	66, 109	sylbird 250	. . . . . . . . . . . 12 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( -. a ~< om -> -. ( a F s ) e. a ) )
111	110	con4d 114	. . . . . . . . . . 11 |- ( ( ph /\ ( a C_ A /\ s C_ ( a X. a ) /\ s We a ) ) -> ( ( a F s ) e. a -> a ~< om ) )
112	48, 56, 111	vtocl 3259	. . . . . . . . . 10 |- ( ( ph /\ ( a C_ A /\ ( W ` Z ) C_ ( a X. a ) /\ ( W ` Z ) We a ) ) -> ( ( a F ( W ` Z ) ) e. a -> a ~< om ) )
113	47, 112	vtoclg 3266	. . . . . . . . 9 |- ( Z e. _V -> ( ph -> ( ( Z F ( W ` Z ) ) e. Z -> Z ~< om ) ) )
114	29, 113	mpcom 38	. . . . . . . 8 |- ( ph -> ( ( Z F ( W ` Z ) ) e. Z -> Z ~< om ) )
115	23, 114	mpd 15	. . . . . . 7 |- ( ph -> Z ~< om )
116		isfinite 8549	. . . . . . 7 |- ( Z e. Fin <-> Z ~< om )
117	115, 116	sylibr 224	. . . . . 6 |- ( ph -> Z e. Fin )
118	5, 13, 14, 15, 16, 17, 18	pwfseqlem2 9481	. . . . . 6 |- ( ( Z e. Fin /\ ( W ` Z ) e. _V ) -> ( Z F ( W ` Z ) ) = ( H ` ( card ` Z ) ) )
119	117, 48, 118	sylancl 694	. . . . 5 |- ( ph -> ( Z F ( W ` Z ) ) = ( H ` ( card ` Z ) ) )
120	119, 23	eqeltrrd 2702	. . . 4 |- ( ph -> ( H ` ( card ` Z ) ) e. Z )
121	4, 12, 24	fpwwe2lem3 9455	. . . . . . . . . 10 |- ( ( ph /\ ( H ` ( card ` Z ) ) e. Z ) -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) F ( ( W ` Z ) i^i ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) X. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) ) = ( H ` ( card ` Z ) ) )
122	120, 121	mpdan 702	. . . . . . . . 9 |- ( ph -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) F ( ( W ` Z ) i^i ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) X. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) ) = ( H ` ( card ` Z ) ) )
123		cnvimass 5485	. . . . . . . . . . . 12 |- ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C_ dom ( W ` Z )
124	27	simprd 479	. . . . . . . . . . . . . 14 |- ( ph -> ( W ` Z ) C_ ( Z X. Z ) )
125		dmss 5323	. . . . . . . . . . . . . 14 |- ( ( W ` Z ) C_ ( Z X. Z ) -> dom ( W ` Z ) C_ dom ( Z X. Z ) )
126	124, 125	syl 17	. . . . . . . . . . . . 13 |- ( ph -> dom ( W ` Z ) C_ dom ( Z X. Z ) )
127		dmxpss 5565	. . . . . . . . . . . . 13 |- dom ( Z X. Z ) C_ Z
128	126, 127	syl6ss 3615	. . . . . . . . . . . 12 |- ( ph -> dom ( W ` Z ) C_ Z )
129	123, 128	syl5ss 3614	. . . . . . . . . . 11 |- ( ph -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C_ Z )
130	117, 129	ssfid 8183	. . . . . . . . . 10 |- ( ph -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. Fin )
131	48	inex1 4799	. . . . . . . . . 10 |- ( ( W ` Z ) i^i ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) X. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) e. _V
132	5, 13, 14, 15, 16, 17, 18	pwfseqlem2 9481	. . . . . . . . . 10 |- ( ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. Fin /\ ( ( W ` Z ) i^i ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) X. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) e. _V ) -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) F ( ( W ` Z ) i^i ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) X. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) ) = ( H ` ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) )
133	130, 131, 132	sylancl 694	. . . . . . . . 9 |- ( ph -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) F ( ( W ` Z ) i^i ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) X. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) ) = ( H ` ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) )
134	122, 133	eqtr3d 2658	. . . . . . . 8 |- ( ph -> ( H ` ( card ` Z ) ) = ( H ` ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) )
135		f1of1 6136	. . . . . . . . . 10 |- ( H : om -1-1-onto-> X -> H : om -1-1-> X )
136	14, 135	syl 17	. . . . . . . . 9 |- ( ph -> H : om -1-1-> X )
137		ficardom 8787	. . . . . . . . . 10 |- ( Z e. Fin -> ( card ` Z ) e. om )
138	117, 137	syl 17	. . . . . . . . 9 |- ( ph -> ( card ` Z ) e. om )
139		ficardom 8787	. . . . . . . . . 10 |- ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. Fin -> ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) e. om )
140	130, 139	syl 17	. . . . . . . . 9 |- ( ph -> ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) e. om )
141		f1fveq 6519	. . . . . . . . 9 |- ( ( H : om -1-1-> X /\ ( ( card ` Z ) e. om /\ ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) e. om ) ) -> ( ( H ` ( card ` Z ) ) = ( H ` ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) <-> ( card ` Z ) = ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) )
142	136, 138, 140, 141	syl12anc 1324	. . . . . . . 8 |- ( ph -> ( ( H ` ( card ` Z ) ) = ( H ` ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) <-> ( card ` Z ) = ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) ) )
143	134, 142	mpbid 222	. . . . . . 7 |- ( ph -> ( card ` Z ) = ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) )
144	143	eqcomd 2628	. . . . . 6 |- ( ph -> ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) = ( card ` Z ) )
145		finnum 8774	. . . . . . . 8 |- ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. Fin -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. dom card )
146	130, 145	syl 17	. . . . . . 7 |- ( ph -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. dom card )
147		finnum 8774	. . . . . . . 8 |- ( Z e. Fin -> Z e. dom card )
148	117, 147	syl 17	. . . . . . 7 |- ( ph -> Z e. dom card )
149		carden2 8813	. . . . . . 7 |- ( ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) e. dom card /\ Z e. dom card ) -> ( ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) = ( card ` Z ) <-> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z ) )
150	146, 148, 149	syl2anc 693	. . . . . 6 |- ( ph -> ( ( card ` ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ) = ( card ` Z ) <-> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z ) )
151	144, 150	mpbid 222	. . . . 5 |- ( ph -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z )
152		dfpss2 3692	. . . . . . . 8 |- ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z <-> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C_ Z /\ -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) = Z ) )
153	152	baib 944	. . . . . . 7 |- ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C_ Z -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z <-> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) = Z ) )
154	129, 153	syl 17	. . . . . 6 |- ( ph -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z <-> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) = Z ) )
155		php3 8146	. . . . . . . . 9 |- ( ( Z e. Fin /\ ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z ) -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~< Z )
156		sdomnen 7984	. . . . . . . . 9 |- ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~< Z -> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z )
157	155, 156	syl 17	. . . . . . . 8 |- ( ( Z e. Fin /\ ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z ) -> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z )
158	157	ex 450	. . . . . . 7 |- ( Z e. Fin -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z -> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z ) )
159	117, 158	syl 17	. . . . . 6 |- ( ph -> ( ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) C. Z -> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z ) )
160	154, 159	sylbird 250	. . . . 5 |- ( ph -> ( -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) = Z -> -. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) ~~ Z ) )
161	151, 160	mt4d 152	. . . 4 |- ( ph -> ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) = Z )
162	120, 161	eleqtrrd 2704	. . 3 |- ( ph -> ( H ` ( card ` Z ) ) e. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) )
163		fvex 6201	. . . 4 |- ( H ` ( card ` Z ) ) e. _V
164	163	eliniseg 5494	. . . 4 |- ( ( H ` ( card ` Z ) ) e. _V -> ( ( H ` ( card ` Z ) ) e. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) <-> ( H ` ( card ` Z ) ) ( W ` Z ) ( H ` ( card ` Z ) ) ) )
165	163, 164	ax-mp 5	. . 3 |- ( ( H ` ( card ` Z ) ) e. ( `' ( W ` Z ) " { ( H ` ( card ` Z ) ) } ) <-> ( H ` ( card ` Z ) ) ( W ` Z ) ( H ` ( card ` Z ) ) )
166	162, 165	sylib 208	. 2 |- ( ph -> ( H ` ( card ` Z ) ) ( W ` Z ) ( H ` ( card ` Z ) ) )
167	26	simprd 479	. . . . 5 |- ( ph -> ( ( W ` Z ) We Z /\ A. b e. Z [. ( `' ( W ` Z ) " { b } ) / v ]. ( v F ( ( W ` Z ) i^i ( v X. v ) ) ) = b ) )
168	167	simpld 475	. . . 4 |- ( ph -> ( W ` Z ) We Z )
169		weso 5105	. . . 4 |- ( ( W ` Z ) We Z -> ( W ` Z ) Or Z )
170	168, 169	syl 17	. . 3 |- ( ph -> ( W ` Z ) Or Z )
171		sonr 5056	. . 3 |- ( ( ( W ` Z ) Or Z /\ ( H ` ( card ` Z ) ) e. Z ) -> -. ( H ` ( card ` Z ) ) ( W ` Z ) ( H ` ( card ` Z ) ) )
172	170, 120, 171	syl2anc 693	. 2 |- ( ph -> -. ( H ` ( card ` Z ) ) ( W ` Z ) ( H ` ( card ` Z ) ) )
173	166, 172	pm2.65i 185	1 |- -. ph

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   |^|cint 4475   U_ciun 4520   class class class wbr 4653   {copab 4712   Or wor 5034   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   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  ax-inf2 8538

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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765

This theorem is referenced by:  pwfseqlem5  9485