Theorem ackbij1lem14 9055

Description: Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )

Assertion

Ref	Expression
ackbij1lem14	|- ( A e. om -> ( F ` { A } ) = suc ( F ` A ) )

Distinct variable groups:   x, F, y   x, A, y

Proof of Theorem ackbij1lem14

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ackbij.f	. . 3 |- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )
2	1	ackbij1lem8 9049	. 2 |- ( A e. om -> ( F ` { A } ) = ( card ` ~P A ) )
3		pweq 4161	. . . . 5 |- ( a = (/) -> ~P a = ~P (/) )
4	3	fveq2d 6195	. . . 4 |- ( a = (/) -> ( card ` ~P a ) = ( card ` ~P (/) ) )
5		fveq2 6191	. . . . 5 |- ( a = (/) -> ( F ` a ) = ( F ` (/) ) )
6		suceq 5790	. . . . 5 |- ( ( F ` a ) = ( F ` (/) ) -> suc ( F ` a ) = suc ( F ` (/) ) )
7	5, 6	syl 17	. . . 4 |- ( a = (/) -> suc ( F ` a ) = suc ( F ` (/) ) )
8	4, 7	eqeq12d 2637	. . 3 |- ( a = (/) -> ( ( card ` ~P a ) = suc ( F ` a ) <-> ( card ` ~P (/) ) = suc ( F ` (/) ) ) )
9		pweq 4161	. . . . 5 |- ( a = b -> ~P a = ~P b )
10	9	fveq2d 6195	. . . 4 |- ( a = b -> ( card ` ~P a ) = ( card ` ~P b ) )
11		fveq2 6191	. . . . 5 |- ( a = b -> ( F ` a ) = ( F ` b ) )
12		suceq 5790	. . . . 5 |- ( ( F ` a ) = ( F ` b ) -> suc ( F ` a ) = suc ( F ` b ) )
13	11, 12	syl 17	. . . 4 |- ( a = b -> suc ( F ` a ) = suc ( F ` b ) )
14	10, 13	eqeq12d 2637	. . 3 |- ( a = b -> ( ( card ` ~P a ) = suc ( F ` a ) <-> ( card ` ~P b ) = suc ( F ` b ) ) )
15		pweq 4161	. . . . 5 |- ( a = suc b -> ~P a = ~P suc b )
16	15	fveq2d 6195	. . . 4 |- ( a = suc b -> ( card ` ~P a ) = ( card ` ~P suc b ) )
17		fveq2 6191	. . . . 5 |- ( a = suc b -> ( F ` a ) = ( F ` suc b ) )
18		suceq 5790	. . . . 5 |- ( ( F ` a ) = ( F ` suc b ) -> suc ( F ` a ) = suc ( F ` suc b ) )
19	17, 18	syl 17	. . . 4 |- ( a = suc b -> suc ( F ` a ) = suc ( F ` suc b ) )
20	16, 19	eqeq12d 2637	. . 3 |- ( a = suc b -> ( ( card ` ~P a ) = suc ( F ` a ) <-> ( card ` ~P suc b ) = suc ( F ` suc b ) ) )
21		pweq 4161	. . . . 5 |- ( a = A -> ~P a = ~P A )
22	21	fveq2d 6195	. . . 4 |- ( a = A -> ( card ` ~P a ) = ( card ` ~P A ) )
23		fveq2 6191	. . . . 5 |- ( a = A -> ( F ` a ) = ( F ` A ) )
24		suceq 5790	. . . . 5 |- ( ( F ` a ) = ( F ` A ) -> suc ( F ` a ) = suc ( F ` A ) )
25	23, 24	syl 17	. . . 4 |- ( a = A -> suc ( F ` a ) = suc ( F ` A ) )
26	22, 25	eqeq12d 2637	. . 3 |- ( a = A -> ( ( card ` ~P a ) = suc ( F ` a ) <-> ( card ` ~P A ) = suc ( F ` A ) ) )
27		df-1o 7560	. . . 4 |- 1o = suc (/)
28		pw0 4343	. . . . . 6 |- ~P (/) = { (/) }
29	28	fveq2i 6194	. . . . 5 |- ( card ` ~P (/) ) = ( card ` { (/) } )
30		0ex 4790	. . . . . 6 |- (/) e. _V
31		cardsn 8795	. . . . . 6 |- ( (/) e. _V -> ( card ` { (/) } ) = 1o )
32	30, 31	ax-mp 5	. . . . 5 |- ( card ` { (/) } ) = 1o
33	29, 32	eqtri 2644	. . . 4 |- ( card ` ~P (/) ) = 1o
34	1	ackbij1lem13 9054	. . . . 5 |- ( F ` (/) ) = (/)
35		suceq 5790	. . . . 5 |- ( ( F ` (/) ) = (/) -> suc ( F ` (/) ) = suc (/) )
36	34, 35	ax-mp 5	. . . 4 |- suc ( F ` (/) ) = suc (/)
37	27, 33, 36	3eqtr4i 2654	. . 3 |- ( card ` ~P (/) ) = suc ( F ` (/) )
38		oveq2 6658	. . . . . 6 |- ( ( card ` ~P b ) = suc ( F ` b ) -> ( ( card ` ~P b ) +o ( card ` ~P b ) ) = ( ( card ` ~P b ) +o suc ( F ` b ) ) )
39	38	adantl 482	. . . . 5 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( ( card ` ~P b ) +o ( card ` ~P b ) ) = ( ( card ` ~P b ) +o suc ( F ` b ) ) )
40		ackbij1lem5 9046	. . . . . 6 |- ( b e. om -> ( card ` ~P suc b ) = ( ( card ` ~P b ) +o ( card ` ~P b ) ) )
41	40	adantr 481	. . . . 5 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( card ` ~P suc b ) = ( ( card ` ~P b ) +o ( card ` ~P b ) ) )
42		df-suc 5729	. . . . . . . . . 10 |- suc b = ( b u. { b } )
43	42	equncomi 3759	. . . . . . . . 9 |- suc b = ( { b } u. b )
44	43	fveq2i 6194	. . . . . . . 8 |- ( F ` suc b ) = ( F ` ( { b } u. b ) )
45		ackbij1lem4 9045	. . . . . . . . . . 11 |- ( b e. om -> { b } e. ( ~P om i^i Fin ) )
46	45	adantr 481	. . . . . . . . . 10 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> { b } e. ( ~P om i^i Fin ) )
47		ackbij1lem3 9044	. . . . . . . . . . 11 |- ( b e. om -> b e. ( ~P om i^i Fin ) )
48	47	adantr 481	. . . . . . . . . 10 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> b e. ( ~P om i^i Fin ) )
49		incom 3805	. . . . . . . . . . . 12 |- ( { b } i^i b ) = ( b i^i { b } )
50		nnord 7073	. . . . . . . . . . . . 13 |- ( b e. om -> Ord b )
51		orddisj 5762	. . . . . . . . . . . . 13 |- ( Ord b -> ( b i^i { b } ) = (/) )
52	50, 51	syl 17	. . . . . . . . . . . 12 |- ( b e. om -> ( b i^i { b } ) = (/) )
53	49, 52	syl5eq 2668	. . . . . . . . . . 11 |- ( b e. om -> ( { b } i^i b ) = (/) )
54	53	adantr 481	. . . . . . . . . 10 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( { b } i^i b ) = (/) )
55	1	ackbij1lem9 9050	. . . . . . . . . 10 |- ( ( { b } e. ( ~P om i^i Fin ) /\ b e. ( ~P om i^i Fin ) /\ ( { b } i^i b ) = (/) ) -> ( F ` ( { b } u. b ) ) = ( ( F ` { b } ) +o ( F ` b ) ) )
56	46, 48, 54, 55	syl3anc 1326	. . . . . . . . 9 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( F ` ( { b } u. b ) ) = ( ( F ` { b } ) +o ( F ` b ) ) )
57	1	ackbij1lem8 9049	. . . . . . . . . . 11 |- ( b e. om -> ( F ` { b } ) = ( card ` ~P b ) )
58	57	adantr 481	. . . . . . . . . 10 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( F ` { b } ) = ( card ` ~P b ) )
59	58	oveq1d 6665	. . . . . . . . 9 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( ( F ` { b } ) +o ( F ` b ) ) = ( ( card ` ~P b ) +o ( F ` b ) ) )
60	56, 59	eqtrd 2656	. . . . . . . 8 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( F ` ( { b } u. b ) ) = ( ( card ` ~P b ) +o ( F ` b ) ) )
61	44, 60	syl5eq 2668	. . . . . . 7 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( F ` suc b ) = ( ( card ` ~P b ) +o ( F ` b ) ) )
62		suceq 5790	. . . . . . 7 |- ( ( F ` suc b ) = ( ( card ` ~P b ) +o ( F ` b ) ) -> suc ( F ` suc b ) = suc ( ( card ` ~P b ) +o ( F ` b ) ) )
63	61, 62	syl 17	. . . . . 6 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> suc ( F ` suc b ) = suc ( ( card ` ~P b ) +o ( F ` b ) ) )
64		nnfi 8153	. . . . . . . . . 10 |- ( b e. om -> b e. Fin )
65		pwfi 8261	. . . . . . . . . 10 |- ( b e. Fin <-> ~P b e. Fin )
66	64, 65	sylib 208	. . . . . . . . 9 |- ( b e. om -> ~P b e. Fin )
67	66	adantr 481	. . . . . . . 8 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ~P b e. Fin )
68		ficardom 8787	. . . . . . . 8 |- ( ~P b e. Fin -> ( card ` ~P b ) e. om )
69	67, 68	syl 17	. . . . . . 7 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( card ` ~P b ) e. om )
70	1	ackbij1lem10 9051	. . . . . . . . 9 |- F : ( ~P om i^i Fin ) --> om
71	70	ffvelrni 6358	. . . . . . . 8 |- ( b e. ( ~P om i^i Fin ) -> ( F ` b ) e. om )
72	48, 71	syl 17	. . . . . . 7 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( F ` b ) e. om )
73		nnasuc 7686	. . . . . . 7 |- ( ( ( card ` ~P b ) e. om /\ ( F ` b ) e. om ) -> ( ( card ` ~P b ) +o suc ( F ` b ) ) = suc ( ( card ` ~P b ) +o ( F ` b ) ) )
74	69, 72, 73	syl2anc 693	. . . . . 6 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( ( card ` ~P b ) +o suc ( F ` b ) ) = suc ( ( card ` ~P b ) +o ( F ` b ) ) )
75	63, 74	eqtr4d 2659	. . . . 5 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> suc ( F ` suc b ) = ( ( card ` ~P b ) +o suc ( F ` b ) ) )
76	39, 41, 75	3eqtr4d 2666	. . . 4 |- ( ( b e. om /\ ( card ` ~P b ) = suc ( F ` b ) ) -> ( card ` ~P suc b ) = suc ( F ` suc b ) )
77	76	ex 450	. . 3 |- ( b e. om -> ( ( card ` ~P b ) = suc ( F ` b ) -> ( card ` ~P suc b ) = suc ( F ` suc b ) ) )
78	8, 14, 20, 26, 37, 77	finds 7092	. 2 |- ( A e. om -> ( card ` ~P A ) = suc ( F ` A ) )
79	2, 78	eqtrd 2656	1 |- ( A e. om -> ( F ` { A } ) = suc ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   Ord word 5722   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   1oc1o 7553   +o coa 7557   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

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-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-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-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by:  ackbij1lem15  9056  ackbij1lem18  9059  ackbij1b  9061