Theorem finxpreclem4 33231

Description: Lemma for ^^ ^^ recursion theorems. (Contributed by ML, 23-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem4.1	|- F = ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )

Assertion

Ref	Expression
finxpreclem4	|- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. N , y >. ) ` N ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` U. N ) )

Distinct variable groups:   n, N, x   U, n, x   y, n, x

Allowed substitution hints:   U( y)   F( x, y, n)   N( y)

Proof of Theorem finxpreclem4

Dummy variable o is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2onn 7720	. . . . . . . 8 |- 2o e. om
2		nnon 7071	. . . . . . . . . . 11 |- ( N e. om -> N e. On )
3		2on 7568	. . . . . . . . . . . . . 14 |- 2o e. On
4		oawordeu 7635	. . . . . . . . . . . . . 14 |- ( ( ( 2o e. On /\ N e. On ) /\ 2o C_ N ) -> E! o e. On ( 2o +o o ) = N )
5	3, 4	mpanl1 716	. . . . . . . . . . . . 13 |- ( ( N e. On /\ 2o C_ N ) -> E! o e. On ( 2o +o o ) = N )
6		riotasbc 6626	. . . . . . . . . . . . 13 |- ( E! o e. On ( 2o +o o ) = N -> [. ( iota_ o e. On ( 2o +o o ) = N ) / o ]. ( 2o +o o ) = N )
7	5, 6	syl 17	. . . . . . . . . . . 12 |- ( ( N e. On /\ 2o C_ N ) -> [. ( iota_ o e. On ( 2o +o o ) = N ) / o ]. ( 2o +o o ) = N )
8		riotaex 6615	. . . . . . . . . . . . . 14 |- ( iota_ o e. On ( 2o +o o ) = N ) e. _V
9		sbceq1g 3988	. . . . . . . . . . . . . 14 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. _V -> ( [. ( iota_ o e. On ( 2o +o o ) = N ) / o ]. ( 2o +o o ) = N <-> [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ ( 2o +o o ) = N ) )
10	8, 9	ax-mp 5	. . . . . . . . . . . . 13 |- ( [. ( iota_ o e. On ( 2o +o o ) = N ) / o ]. ( 2o +o o ) = N <-> [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ ( 2o +o o ) = N )
11		csbov2g 6691	. . . . . . . . . . . . . . . 16 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. _V -> [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ ( 2o +o o ) = ( 2o +o [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ o ) )
12	8, 11	ax-mp 5	. . . . . . . . . . . . . . 15 |- [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ ( 2o +o o ) = ( 2o +o [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ o )
13		csbvarg 4003	. . . . . . . . . . . . . . . . 17 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. _V -> [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ o = ( iota_ o e. On ( 2o +o o ) = N ) )
14	8, 13	ax-mp 5	. . . . . . . . . . . . . . . 16 |- [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ o = ( iota_ o e. On ( 2o +o o ) = N )
15	14	oveq2i 6661	. . . . . . . . . . . . . . 15 |- ( 2o +o [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ o ) = ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) )
16	12, 15	eqtri 2644	. . . . . . . . . . . . . 14 |- [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ ( 2o +o o ) = ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) )
17	16	eqeq1i 2627	. . . . . . . . . . . . 13 |- ( [_ ( iota_ o e. On ( 2o +o o ) = N ) / o ]_ ( 2o +o o ) = N <-> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) = N )
18	10, 17	bitri 264	. . . . . . . . . . . 12 |- ( [. ( iota_ o e. On ( 2o +o o ) = N ) / o ]. ( 2o +o o ) = N <-> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) = N )
19	7, 18	sylib 208	. . . . . . . . . . 11 |- ( ( N e. On /\ 2o C_ N ) -> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) = N )
20	2, 19	sylan 488	. . . . . . . . . 10 |- ( ( N e. om /\ 2o C_ N ) -> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) = N )
21		simpl 473	. . . . . . . . . 10 |- ( ( N e. om /\ 2o C_ N ) -> N e. om )
22	20, 21	eqeltrd 2701	. . . . . . . . 9 |- ( ( N e. om /\ 2o C_ N ) -> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om )
23		riotacl 6625	. . . . . . . . . . 11 |- ( E! o e. On ( 2o +o o ) = N -> ( iota_ o e. On ( 2o +o o ) = N ) e. On )
24		riotaund 6647	. . . . . . . . . . . 12 |- ( -. E! o e. On ( 2o +o o ) = N -> ( iota_ o e. On ( 2o +o o ) = N ) = (/) )
25		0elon 5778	. . . . . . . . . . . 12 |- (/) e. On
26	24, 25	syl6eqel 2709	. . . . . . . . . . 11 |- ( -. E! o e. On ( 2o +o o ) = N -> ( iota_ o e. On ( 2o +o o ) = N ) e. On )
27	23, 26	pm2.61i 176	. . . . . . . . . 10 |- ( iota_ o e. On ( 2o +o o ) = N ) e. On
28		nnarcl 7696	. . . . . . . . . . . 12 |- ( ( 2o e. On /\ ( iota_ o e. On ( 2o +o o ) = N ) e. On ) -> ( ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om <-> ( 2o e. om /\ ( iota_ o e. On ( 2o +o o ) = N ) e. om ) ) )
29	3, 28	mpan 706	. . . . . . . . . . 11 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. On -> ( ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om <-> ( 2o e. om /\ ( iota_ o e. On ( 2o +o o ) = N ) e. om ) ) )
30	1	biantrur 527	. . . . . . . . . . 11 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. om <-> ( 2o e. om /\ ( iota_ o e. On ( 2o +o o ) = N ) e. om ) )
31	29, 30	syl6bbr 278	. . . . . . . . . 10 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. On -> ( ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om <-> ( iota_ o e. On ( 2o +o o ) = N ) e. om ) )
32	27, 31	ax-mp 5	. . . . . . . . 9 |- ( ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om <-> ( iota_ o e. On ( 2o +o o ) = N ) e. om )
33	22, 32	sylib 208	. . . . . . . 8 |- ( ( N e. om /\ 2o C_ N ) -> ( iota_ o e. On ( 2o +o o ) = N ) e. om )
34		nnacom 7697	. . . . . . . 8 |- ( ( 2o e. om /\ ( iota_ o e. On ( 2o +o o ) = N ) e. om ) -> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) = ( ( iota_ o e. On ( 2o +o o ) = N ) +o 2o ) )
35	1, 33, 34	sylancr 695	. . . . . . 7 |- ( ( N e. om /\ 2o C_ N ) -> ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) = ( ( iota_ o e. On ( 2o +o o ) = N ) +o 2o ) )
36		df-2o 7561	. . . . . . . . 9 |- 2o = suc 1o
37	36	oveq2i 6661	. . . . . . . 8 |- ( ( iota_ o e. On ( 2o +o o ) = N ) +o 2o ) = ( ( iota_ o e. On ( 2o +o o ) = N ) +o suc 1o )
38		1onn 7719	. . . . . . . . 9 |- 1o e. om
39		nnasuc 7686	. . . . . . . . 9 |- ( ( ( iota_ o e. On ( 2o +o o ) = N ) e. om /\ 1o e. om ) -> ( ( iota_ o e. On ( 2o +o o ) = N ) +o suc 1o ) = suc ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) )
40	33, 38, 39	sylancl 694	. . . . . . . 8 |- ( ( N e. om /\ 2o C_ N ) -> ( ( iota_ o e. On ( 2o +o o ) = N ) +o suc 1o ) = suc ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) )
41	37, 40	syl5eq 2668	. . . . . . 7 |- ( ( N e. om /\ 2o C_ N ) -> ( ( iota_ o e. On ( 2o +o o ) = N ) +o 2o ) = suc ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) )
42	35, 20, 41	3eqtr3d 2664	. . . . . 6 |- ( ( N e. om /\ 2o C_ N ) -> N = suc ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) )
43	2	adantr 481	. . . . . . 7 |- ( ( N e. om /\ 2o C_ N ) -> N e. On )
44		sucidg 5803	. . . . . . . . . . . 12 |- ( 1o e. om -> 1o e. suc 1o )
45	38, 44	ax-mp 5	. . . . . . . . . . 11 |- 1o e. suc 1o
46	45, 36	eleqtrri 2700	. . . . . . . . . 10 |- 1o e. 2o
47		ssel 3597	. . . . . . . . . 10 |- ( 2o C_ N -> ( 1o e. 2o -> 1o e. N ) )
48	46, 47	mpi 20	. . . . . . . . 9 |- ( 2o C_ N -> 1o e. N )
49		ne0i 3921	. . . . . . . . 9 |- ( 1o e. N -> N =/= (/) )
50	48, 49	syl 17	. . . . . . . 8 |- ( 2o C_ N -> N =/= (/) )
51	50	adantl 482	. . . . . . 7 |- ( ( N e. om /\ 2o C_ N ) -> N =/= (/) )
52		nnlim 7078	. . . . . . . 8 |- ( N e. om -> -. Lim N )
53	52	adantr 481	. . . . . . 7 |- ( ( N e. om /\ 2o C_ N ) -> -. Lim N )
54		onsucuni3 33215	. . . . . . 7 |- ( ( N e. On /\ N =/= (/) /\ -. Lim N ) -> N = suc U. N )
55	43, 51, 53, 54	syl3anc 1326	. . . . . 6 |- ( ( N e. om /\ 2o C_ N ) -> N = suc U. N )
56		nnacom 7697	. . . . . . . 8 |- ( ( ( iota_ o e. On ( 2o +o o ) = N ) e. om /\ 1o e. om ) -> ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) = ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) )
57	33, 38, 56	sylancl 694	. . . . . . 7 |- ( ( N e. om /\ 2o C_ N ) -> ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) = ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) )
58		suceq 5790	. . . . . . 7 |- ( ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) = ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) -> suc ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) = suc ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) )
59	57, 58	syl 17	. . . . . 6 |- ( ( N e. om /\ 2o C_ N ) -> suc ( ( iota_ o e. On ( 2o +o o ) = N ) +o 1o ) = suc ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) )
60	42, 55, 59	3eqtr3d 2664	. . . . 5 |- ( ( N e. om /\ 2o C_ N ) -> suc U. N = suc ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) )
61		ordom 7074	. . . . . . . . 9 |- Ord om
62		ordelss 5739	. . . . . . . . 9 |- ( ( Ord om /\ N e. om ) -> N C_ om )
63	61, 62	mpan 706	. . . . . . . 8 |- ( N e. om -> N C_ om )
64		nnfi 8153	. . . . . . . 8 |- ( N e. om -> N e. Fin )
65		nnunifi 8211	. . . . . . . 8 |- ( ( N C_ om /\ N e. Fin ) -> U. N e. om )
66	63, 64, 65	syl2anc 693	. . . . . . 7 |- ( N e. om -> U. N e. om )
67	66	adantr 481	. . . . . 6 |- ( ( N e. om /\ 2o C_ N ) -> U. N e. om )
68		nnacl 7691	. . . . . . 7 |- ( ( 1o e. om /\ ( iota_ o e. On ( 2o +o o ) = N ) e. om ) -> ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om )
69	38, 33, 68	sylancr 695	. . . . . 6 |- ( ( N e. om /\ 2o C_ N ) -> ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om )
70		peano4 7088	. . . . . 6 |- ( ( U. N e. om /\ ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) e. om ) -> ( suc U. N = suc ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) <-> U. N = ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) )
71	67, 69, 70	syl2anc 693	. . . . 5 |- ( ( N e. om /\ 2o C_ N ) -> ( suc U. N = suc ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) <-> U. N = ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) )
72	60, 71	mpbid 222	. . . 4 |- ( ( N e. om /\ 2o C_ N ) -> U. N = ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) )
73	72	fveq2d 6195	. . 3 |- ( ( N e. om /\ 2o C_ N ) -> ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` U. N ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) )
74	73	adantr 481	. 2 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` U. N ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) )
75	33	adantr 481	. . 3 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( iota_ o e. On ( 2o +o o ) = N ) e. om )
76		finxpreclem4.1	. . . . . . 7 |- F = ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) )
77	76	finxpreclem3 33230	. . . . . 6 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> <. U. N , ( 1st ` y ) >. = ( F ` <. N , y >. ) )
78		df-1o 7560	. . . . . . . 8 |- 1o = suc (/)
79	78	fveq2i 6194	. . . . . . 7 |- ( rec ( F , <. N , y >. ) ` 1o ) = ( rec ( F , <. N , y >. ) ` suc (/) )
80		rdgsuc 7520	. . . . . . . 8 |- ( (/) e. On -> ( rec ( F , <. N , y >. ) ` suc (/) ) = ( F ` ( rec ( F , <. N , y >. ) ` (/) ) ) )
81	25, 80	ax-mp 5	. . . . . . 7 |- ( rec ( F , <. N , y >. ) ` suc (/) ) = ( F ` ( rec ( F , <. N , y >. ) ` (/) ) )
82		opex 4932	. . . . . . . . 9 |- <. N , y >. e. _V
83	82	rdg0 7517	. . . . . . . 8 |- ( rec ( F , <. N , y >. ) ` (/) ) = <. N , y >.
84	83	fveq2i 6194	. . . . . . 7 |- ( F ` ( rec ( F , <. N , y >. ) ` (/) ) ) = ( F ` <. N , y >. )
85	79, 81, 84	3eqtri 2648	. . . . . 6 |- ( rec ( F , <. N , y >. ) ` 1o ) = ( F ` <. N , y >. )
86	77, 85	syl6reqr 2675	. . . . 5 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. N , y >. ) ` 1o ) = <. U. N , ( 1st ` y ) >. )
87	86	fveq2d 6195	. . . 4 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( F ` ( rec ( F , <. N , y >. ) ` 1o ) ) = ( F ` <. U. N , ( 1st ` y ) >. ) )
88		2on0 7569	. . . . . 6 |- 2o =/= (/)
89		nnlim 7078	. . . . . . 7 |- ( 2o e. om -> -. Lim 2o )
90	1, 89	ax-mp 5	. . . . . 6 |- -. Lim 2o
91		rdgsucuni 33217	. . . . . 6 |- ( ( 2o e. On /\ 2o =/= (/) /\ -. Lim 2o ) -> ( rec ( F , <. N , y >. ) ` 2o ) = ( F ` ( rec ( F , <. N , y >. ) ` U. 2o ) ) )
92	3, 88, 90, 91	mp3an 1424	. . . . 5 |- ( rec ( F , <. N , y >. ) ` 2o ) = ( F ` ( rec ( F , <. N , y >. ) ` U. 2o ) )
93		1oequni2o 33216	. . . . . . 7 |- 1o = U. 2o
94	93	fveq2i 6194	. . . . . 6 |- ( rec ( F , <. N , y >. ) ` 1o ) = ( rec ( F , <. N , y >. ) ` U. 2o )
95	94	fveq2i 6194	. . . . 5 |- ( F ` ( rec ( F , <. N , y >. ) ` 1o ) ) = ( F ` ( rec ( F , <. N , y >. ) ` U. 2o ) )
96	92, 95	eqtr4i 2647	. . . 4 |- ( rec ( F , <. N , y >. ) ` 2o ) = ( F ` ( rec ( F , <. N , y >. ) ` 1o ) )
97	78	fveq2i 6194	. . . . 5 |- ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` 1o ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` suc (/) )
98		rdgsuc 7520	. . . . . 6 |- ( (/) e. On -> ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` suc (/) ) = ( F ` ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` (/) ) ) )
99	25, 98	ax-mp 5	. . . . 5 |- ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` suc (/) ) = ( F ` ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` (/) ) )
100		opex 4932	. . . . . . 7 |- <. U. N , ( 1st ` y ) >. e. _V
101	100	rdg0 7517	. . . . . 6 |- ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` (/) ) = <. U. N , ( 1st ` y ) >.
102	101	fveq2i 6194	. . . . 5 |- ( F ` ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` (/) ) ) = ( F ` <. U. N , ( 1st ` y ) >. )
103	97, 99, 102	3eqtri 2648	. . . 4 |- ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` 1o ) = ( F ` <. U. N , ( 1st ` y ) >. )
104	87, 96, 103	3eqtr4g 2681	. . 3 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. N , y >. ) ` 2o ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` 1o ) )
105		1on 7567	. . . 4 |- 1o e. On
106		rdgeqoa 33218	. . . 4 |- ( ( 2o e. On /\ 1o e. On /\ ( iota_ o e. On ( 2o +o o ) = N ) e. om ) -> ( ( rec ( F , <. N , y >. ) ` 2o ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` 1o ) -> ( rec ( F , <. N , y >. ) ` ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) ) )
107	3, 105, 106	mp3an12 1414	. . 3 |- ( ( iota_ o e. On ( 2o +o o ) = N ) e. om -> ( ( rec ( F , <. N , y >. ) ` 2o ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` 1o ) -> ( rec ( F , <. N , y >. ) ` ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) ) )
108	75, 104, 107	sylc 65	. 2 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. N , y >. ) ` ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` ( 1o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) )
109	20	fveq2d 6195	. . 3 |- ( ( N e. om /\ 2o C_ N ) -> ( rec ( F , <. N , y >. ) ` ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) = ( rec ( F , <. N , y >. ) ` N ) )
110	109	adantr 481	. 2 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. N , y >. ) ` ( 2o +o ( iota_ o e. On ( 2o +o o ) = N ) ) ) = ( rec ( F , <. N , y >. ) ` N ) )
111	74, 108, 110	3eqtr2rd 2663	1 |- ( ( ( N e. om /\ 2o C_ N ) /\ y e. ( _V X. U ) ) -> ( rec ( F , <. N , y >. ) ` N ) = ( rec ( F , <. U. N , ( 1st ` y ) >. ) ` U. N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E!wreu 2914   _Vcvv 3200   [.wsbc 3435   [_csb 3533   C_ wss 3574   (/)c0 3915   ifcif 4086   <.cop 4183   U.cuni 4436   X. cxp 5112   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   reccrdg 7505   1oc1o 7553   2oc2o 7554   +o coa 7557   Fincfn 7955

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-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  finxpsuclem  33234