Theorem tz7.44-2 7125

Description: The value of F at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypotheses

Ref	Expression
tz7.44.1	|- G = ( x e. _V |-> if ( x = (/) , A , if ( Lim dom x , U. ran x , ( H ` ( x ` U. dom x ) ) ) ) )
tz7.44.2	|- ( y e. X -> ( F ` y ) = ( G ` ( F |` y ) ) )
tz7.44.3	|- ( y e. X -> ( F |` y ) e. _V )
tz7.44.4	|- F Fn X
tz7.44.5	|- Ord X

Assertion

Ref	Expression
tz7.44-2	|- ( suc B e. X -> ( F ` suc B ) = ( H ` ( F ` B ) ) )

Distinct variable groups:   x, A   x, y, B   x, F, y   y, G   x, H   y, X

Allowed substitution hints:   A( y)   G( x)   H( y)   X( x)

Proof of Theorem tz7.44-2

Step	Hyp	Ref	Expression
1		fveq2 5865	. . . 4 |- ( y = suc B -> ( F ` y ) = ( F ` suc B ) )
2		reseq2 5100	. . . . 5 |- ( y = suc B -> ( F |` y ) = ( F |` suc B ) )
3	2	fveq2d 5869	. . . 4 |- ( y = suc B -> ( G ` ( F |` y ) ) = ( G ` ( F |` suc B ) ) )
4	1, 3	eqeq12d 2466	. . 3 |- ( y = suc B -> ( ( F ` y ) = ( G ` ( F |` y ) ) <-> ( F ` suc B ) = ( G ` ( F |` suc B ) ) ) )
5		tz7.44.2	. . 3 |- ( y e. X -> ( F ` y ) = ( G ` ( F |` y ) ) )
6	4, 5	vtoclga 3113	. 2 |- ( suc B e. X -> ( F ` suc B ) = ( G ` ( F |` suc B ) ) )
7	2	eleq1d 2513	. . . 4 |- ( y = suc B -> ( ( F |` y ) e. _V <-> ( F |` suc B ) e. _V ) )
8		tz7.44.3	. . . 4 |- ( y e. X -> ( F |` y ) e. _V )
9	7, 8	vtoclga 3113	. . 3 |- ( suc B e. X -> ( F |` suc B ) e. _V )
10		noel 3735	. . . . . . 7 |- -. B e. (/)
11		dmeq 5035	. . . . . . . . 9 |- ( ( F |` suc B ) = (/) -> dom ( F |` suc B ) = dom (/) )
12		dm0 5048	. . . . . . . . 9 |- dom (/) = (/)
13	11, 12	syl6eq 2501	. . . . . . . 8 |- ( ( F |` suc B ) = (/) -> dom ( F |` suc B ) = (/) )
14		tz7.44.5	. . . . . . . . . . . . 13 |- Ord X
15		ordsson 6616	. . . . . . . . . . . . 13 |- ( Ord X -> X C_ On )
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 |- X C_ On
17		ordtr 5437	. . . . . . . . . . . . . 14 |- ( Ord X -> Tr X )
18	14, 17	ax-mp 5	. . . . . . . . . . . . 13 |- Tr X
19		trsuc 5507	. . . . . . . . . . . . 13 |- ( ( Tr X /\ suc B e. X ) -> B e. X )
20	18, 19	mpan 676	. . . . . . . . . . . 12 |- ( suc B e. X -> B e. X )
21	16, 20	sseldi 3430	. . . . . . . . . . 11 |- ( suc B e. X -> B e. On )
22		sucidg 5501	. . . . . . . . . . 11 |- ( B e. On -> B e. suc B )
23	21, 22	syl 17	. . . . . . . . . 10 |- ( suc B e. X -> B e. suc B )
24		dmres 5125	. . . . . . . . . . 11 |- dom ( F |` suc B ) = ( suc B i^i dom F )
25		ordelss 5439	. . . . . . . . . . . . . 14 |- ( ( Ord X /\ suc B e. X ) -> suc B C_ X )
26	14, 25	mpan 676	. . . . . . . . . . . . 13 |- ( suc B e. X -> suc B C_ X )
27		tz7.44.4	. . . . . . . . . . . . . 14 |- F Fn X
28		fndm 5675	. . . . . . . . . . . . . 14 |- ( F Fn X -> dom F = X )
29	27, 28	ax-mp 5	. . . . . . . . . . . . 13 |- dom F = X
30	26, 29	syl6sseqr 3479	. . . . . . . . . . . 12 |- ( suc B e. X -> suc B C_ dom F )
31		df-ss 3418	. . . . . . . . . . . 12 |- ( suc B C_ dom F <-> ( suc B i^i dom F ) = suc B )
32	30, 31	sylib 200	. . . . . . . . . . 11 |- ( suc B e. X -> ( suc B i^i dom F ) = suc B )
33	24, 32	syl5eq 2497	. . . . . . . . . 10 |- ( suc B e. X -> dom ( F |` suc B ) = suc B )
34	23, 33	eleqtrrd 2532	. . . . . . . . 9 |- ( suc B e. X -> B e. dom ( F |` suc B ) )
35		eleq2 2518	. . . . . . . . 9 |- ( dom ( F |` suc B ) = (/) -> ( B e. dom ( F |` suc B ) <-> B e. (/) ) )
36	34, 35	syl5ibcom 224	. . . . . . . 8 |- ( suc B e. X -> ( dom ( F |` suc B ) = (/) -> B e. (/) ) )
37	13, 36	syl5 33	. . . . . . 7 |- ( suc B e. X -> ( ( F |` suc B ) = (/) -> B e. (/) ) )
38	10, 37	mtoi 182	. . . . . 6 |- ( suc B e. X -> -. ( F |` suc B ) = (/) )
39	38	iffalsed 3892	. . . . 5 |- ( suc B e. X -> if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) = if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) )
40		nlimsucg 6669	. . . . . . . 8 |- ( B e. On -> -. Lim suc B )
41	21, 40	syl 17	. . . . . . 7 |- ( suc B e. X -> -. Lim suc B )
42		limeq 5435	. . . . . . . 8 |- ( dom ( F |` suc B ) = suc B -> ( Lim dom ( F |` suc B ) <-> Lim suc B ) )
43	33, 42	syl 17	. . . . . . 7 |- ( suc B e. X -> ( Lim dom ( F |` suc B ) <-> Lim suc B ) )
44	41, 43	mtbird 303	. . . . . 6 |- ( suc B e. X -> -. Lim dom ( F |` suc B ) )
45	44	iffalsed 3892	. . . . 5 |- ( suc B e. X -> if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) = ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) )
46	33	unieqd 4208	. . . . . . . . 9 |- ( suc B e. X -> U. dom ( F |` suc B ) = U. suc B )
47		eloni 5433	. . . . . . . . . . 11 |- ( B e. On -> Ord B )
48		ordunisuc 6659	. . . . . . . . . . 11 |- ( Ord B -> U. suc B = B )
49	47, 48	syl 17	. . . . . . . . . 10 |- ( B e. On -> U. suc B = B )
50	21, 49	syl 17	. . . . . . . . 9 |- ( suc B e. X -> U. suc B = B )
51	46, 50	eqtrd 2485	. . . . . . . 8 |- ( suc B e. X -> U. dom ( F |` suc B ) = B )
52	51	fveq2d 5869	. . . . . . 7 |- ( suc B e. X -> ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) = ( ( F |` suc B ) ` B ) )
53		fvres 5879	. . . . . . . 8 |- ( B e. suc B -> ( ( F |` suc B ) ` B ) = ( F ` B ) )
54	23, 53	syl 17	. . . . . . 7 |- ( suc B e. X -> ( ( F |` suc B ) ` B ) = ( F ` B ) )
55	52, 54	eqtrd 2485	. . . . . 6 |- ( suc B e. X -> ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) = ( F ` B ) )
56	55	fveq2d 5869	. . . . 5 |- ( suc B e. X -> ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) = ( H ` ( F ` B ) ) )
57	39, 45, 56	3eqtrd 2489	. . . 4 |- ( suc B e. X -> if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) = ( H ` ( F ` B ) ) )
58		fvex 5875	. . . 4 |- ( H ` ( F ` B ) ) e. _V
59	57, 58	syl6eqel 2537	. . 3 |- ( suc B e. X -> if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) e. _V )
60		eqeq1 2455	. . . . 5 |- ( x = ( F |` suc B ) -> ( x = (/) <-> ( F |` suc B ) = (/) ) )
61		dmeq 5035	. . . . . . 7 |- ( x = ( F |` suc B ) -> dom x = dom ( F |` suc B ) )
62		limeq 5435	. . . . . . 7 |- ( dom x = dom ( F |` suc B ) -> ( Lim dom x <-> Lim dom ( F |` suc B ) ) )
63	61, 62	syl 17	. . . . . 6 |- ( x = ( F |` suc B ) -> ( Lim dom x <-> Lim dom ( F |` suc B ) ) )
64		rneq 5060	. . . . . . 7 |- ( x = ( F |` suc B ) -> ran x = ran ( F |` suc B ) )
65	64	unieqd 4208	. . . . . 6 |- ( x = ( F |` suc B ) -> U. ran x = U. ran ( F |` suc B ) )
66		fveq1 5864	. . . . . . . 8 |- ( x = ( F |` suc B ) -> ( x ` U. dom x ) = ( ( F |` suc B ) ` U. dom x ) )
67	61	unieqd 4208	. . . . . . . . 9 |- ( x = ( F |` suc B ) -> U. dom x = U. dom ( F |` suc B ) )
68	67	fveq2d 5869	. . . . . . . 8 |- ( x = ( F |` suc B ) -> ( ( F |` suc B ) ` U. dom x ) = ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) )
69	66, 68	eqtrd 2485	. . . . . . 7 |- ( x = ( F |` suc B ) -> ( x ` U. dom x ) = ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) )
70	69	fveq2d 5869	. . . . . 6 |- ( x = ( F |` suc B ) -> ( H ` ( x ` U. dom x ) ) = ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) )
71	63, 65, 70	ifbieq12d 3908	. . . . 5 |- ( x = ( F |` suc B ) -> if ( Lim dom x , U. ran x , ( H ` ( x ` U. dom x ) ) ) = if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) )
72	60, 71	ifbieq2d 3906	. . . 4 |- ( x = ( F |` suc B ) -> if ( x = (/) , A , if ( Lim dom x , U. ran x , ( H ` ( x ` U. dom x ) ) ) ) = if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) )
73		tz7.44.1	. . . 4 |- G = ( x e. _V |-> if ( x = (/) , A , if ( Lim dom x , U. ran x , ( H ` ( x ` U. dom x ) ) ) ) )
74	72, 73	fvmptg 5946	. . 3 |- ( ( ( F |` suc B ) e. _V /\ if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) e. _V ) -> ( G ` ( F |` suc B ) ) = if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) )
75	9, 59, 74	syl2anc 667	. 2 |- ( suc B e. X -> ( G ` ( F |` suc B ) ) = if ( ( F |` suc B ) = (/) , A , if ( Lim dom ( F |` suc B ) , U. ran ( F |` suc B ) , ( H ` ( ( F |` suc B ) ` U. dom ( F |` suc B ) ) ) ) ) )
76	6, 75, 57	3eqtrd 2489	1 |- ( suc B e. X -> ( F ` suc B ) = ( H ` ( F ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   = wceq 1444   e. wcel 1887   _Vcvv 3045   i^i cin 3403   C_ wss 3404   (/)c0 3731   ifcif 3881   U.cuni 4198   |-> cmpt 4461   Tr wtr 4497   dom cdm 4834   ran crn 4835   |` cres 4836   Ord word 5422   Oncon0 5423   Lim wlim 5424   suc csuc 5425   Fn wfn 5577   ` cfv 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590

This theorem is referenced by:  rdgsucg  7141