Theorem frecsuclem3 5990

Description: Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis

Ref	Expression
frecsuclem1.h	|- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )

Assertion

Ref	Expression
frecsuclem3	|- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )

Distinct variable groups:   A, g, m, x, z   B, g, m, x, z   g, F, m, x, z   g, G, m, x, z   g, V, m, x

Allowed substitution hint:   V( z)

Proof of Theorem frecsuclem3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2040	. . . . . . . . . . . . 13 |- recs ( G ) = recs ( G )
2		frecsuclem1.h	. . . . . . . . . . . . . 14 |- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
3	2	frectfr 5985	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )
4	1, 3	tfri1d 5949	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> recs ( G ) Fn On )
5		fnfun 4996	. . . . . . . . . . . 12 |- (recs ( G ) Fn On -> Fun recs ( G ) )
6	4, 5	syl 14	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> Fun recs ( G ) )
7	6	3adant3 924	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> Fun recs ( G ) )
8		peano2 4318	. . . . . . . . . . 11 |- ( B e. om -> suc B e. om )
9	8	3ad2ant3 927	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> suc B e. om )
10		resfunexg 5382	. . . . . . . . . 10 |- ( ( Fun recs ( G ) /\ suc B e. om ) -> (recs ( G ) |` suc B ) e. _V )
11	7, 9, 10	syl2anc 391	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (recs ( G ) |` suc B ) e. _V )
12		simp1 904	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> A. z ( F ` z ) e. _V )
13		simp2 905	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> A e. V )
14	11, 12, 13	frecabex 5984	. . . . . . . 8 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } e. _V )
15		dmeq 4535	. . . . . . . . . . . . . . . . 17 |- ( g = (recs ( G ) |` suc B ) -> dom g = dom (recs ( G ) |` suc B ) )
16	15	eqeq1d 2048	. . . . . . . . . . . . . . . 16 |- ( g = (recs ( G ) |` suc B ) -> ( dom g = suc m <-> dom (recs ( G ) |` suc B ) = suc m ) )
17		fveq1 5177	. . . . . . . . . . . . . . . . . 18 |- ( g = (recs ( G ) |` suc B ) -> ( g ` m ) = ( (recs ( G ) |` suc B ) ` m ) )
18	17	fveq2d 5182	. . . . . . . . . . . . . . . . 17 |- ( g = (recs ( G ) |` suc B ) -> ( F ` ( g ` m ) ) = ( F ` ( (recs ( G ) |` suc B ) ` m ) ) )
19	18	eleq2d 2107	. . . . . . . . . . . . . . . 16 |- ( g = (recs ( G ) |` suc B ) -> ( x e. ( F ` ( g ` m ) ) <-> x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) )
20	16, 19	anbi12d 442	. . . . . . . . . . . . . . 15 |- ( g = (recs ( G ) |` suc B ) -> ( ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) )
21	20	rexbidv 2327	. . . . . . . . . . . . . 14 |- ( g = (recs ( G ) |` suc B ) -> ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) )
22	15	eqeq1d 2048	. . . . . . . . . . . . . . 15 |- ( g = (recs ( G ) |` suc B ) -> ( dom g = (/) <-> dom (recs ( G ) |` suc B ) = (/) ) )
23	22	anbi1d 438	. . . . . . . . . . . . . 14 |- ( g = (recs ( G ) |` suc B ) -> ( ( dom g = (/) /\ x e. A ) <-> ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) )
24	21, 23	orbi12d 707	. . . . . . . . . . . . 13 |- ( g = (recs ( G ) |` suc B ) -> ( ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) <-> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) ) )
25	24	abbidv 2155	. . . . . . . . . . . 12 |- ( g = (recs ( G ) |` suc B ) -> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } )
26	25, 2	fvmptg 5248	. . . . . . . . . . 11 |- ( ( (recs ( G ) |` suc B ) e. _V /\ { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } e. _V ) -> ( G ` (recs ( G ) |` suc B ) ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } )
27	26	ex 108	. . . . . . . . . 10 |- ( (recs ( G ) |` suc B ) e. _V -> ( { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } e. _V -> ( G ` (recs ( G ) |` suc B ) ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } ) )
28	11, 27	syl 14	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } e. _V -> ( G ` (recs ( G ) |` suc B ) ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } ) )
29	2	frecsuclem1 5987	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( G ` (recs ( G ) |` suc B ) ) )
30	29	eqeq1d 2048	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( (frec ( F , A ) ` suc B ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } <-> ( G ` (recs ( G ) |` suc B ) ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } ) )
31	28, 30	sylibrd 158	. . . . . . . 8 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } e. _V -> (frec ( F , A ) ` suc B ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } ) )
32	14, 31	mpd 13	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = { x | ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) } )
33	32	abeq2d 2150	. . . . . 6 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( x e. (frec ( F , A ) ` suc B ) <-> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) ) )
34	2	frecsuclemdm 5988	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> dom (recs ( G ) |` suc B ) = suc B )
35		peano3 4319	. . . . . . . . . . . 12 |- ( B e. om -> suc B =/= (/) )
36	35	3ad2ant3 927	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> suc B =/= (/) )
37	34, 36	eqnetrd 2229	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> dom (recs ( G ) |` suc B ) =/= (/) )
38	37	neneqd 2226	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> -. dom (recs ( G ) |` suc B ) = (/) )
39	38	intnanrd 841	. . . . . . . 8 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> -. ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) )
40		biorf 663	. . . . . . . 8 |- ( -. ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) -> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) <-> ( ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) \/ E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) ) )
41	39, 40	syl 14	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) <-> ( ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) \/ E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) ) )
42		orcom 647	. . . . . . 7 |- ( ( ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) \/ E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) <-> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) )
43	41, 42	syl6bb 185	. . . . . 6 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) <-> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) \/ ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) ) )
44	34	eqeq1d 2048	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( dom (recs ( G ) |` suc B ) = suc m <-> suc B = suc m ) )
45		vex 2560	. . . . . . . . . . . 12 |- m e. _V
46		suc11g 4281	. . . . . . . . . . . 12 |- ( ( B e. om /\ m e. _V ) -> ( suc B = suc m <-> B = m ) )
47	45, 46	mpan2 401	. . . . . . . . . . 11 |- ( B e. om -> ( suc B = suc m <-> B = m ) )
48	47	3ad2ant3 927	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( suc B = suc m <-> B = m ) )
49	44, 48	bitrd 177	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( dom (recs ( G ) |` suc B ) = suc m <-> B = m ) )
50		eqcom 2042	. . . . . . . . 9 |- ( B = m <-> m = B )
51	49, 50	syl6bb 185	. . . . . . . 8 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( dom (recs ( G ) |` suc B ) = suc m <-> m = B ) )
52	51	anbi1d 438	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) <-> ( m = B /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) )
53	52	rexbidv 2327	. . . . . 6 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( E. m e. om ( dom (recs ( G ) |` suc B ) = suc m /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) <-> E. m e. om ( m = B /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) )
54	33, 43, 53	3bitr2d 205	. . . . 5 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( x e. (frec ( F , A ) ` suc B ) <-> E. m e. om ( m = B /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) ) )
55		fveq2 5178	. . . . . . . 8 |- ( m = B -> ( (recs ( G ) |` suc B ) ` m ) = ( (recs ( G ) |` suc B ) ` B ) )
56	55	fveq2d 5182	. . . . . . 7 |- ( m = B -> ( F ` ( (recs ( G ) |` suc B ) ` m ) ) = ( F ` ( (recs ( G ) |` suc B ) ` B ) ) )
57	56	eleq2d 2107	. . . . . 6 |- ( m = B -> ( x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) <-> x e. ( F ` ( (recs ( G ) |` suc B ) ` B ) ) ) )
58	57	ceqsrexbv 2675	. . . . 5 |- ( E. m e. om ( m = B /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) <-> ( B e. om /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` B ) ) ) )
59	54, 58	syl6bb 185	. . . 4 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( x e. (frec ( F , A ) ` suc B ) <-> ( B e. om /\ x e. ( F ` ( (recs ( G ) |` suc B ) ` B ) ) ) ) )
60	59	3anibar 1072	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( x e. (frec ( F , A ) ` suc B ) <-> x e. ( F ` ( (recs ( G ) |` suc B ) ` B ) ) ) )
61	60	eqrdv 2038	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` ( (recs ( G ) |` suc B ) ` B ) ) )
62	2	frecsuclem2 5989	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( (recs ( G ) |` suc B ) ` B ) = (frec ( F , A ) ` B ) )
63	62	fveq2d 5182	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( F ` ( (recs ( G ) |` suc B ) ` B ) ) = ( F ` (frec ( F , A ) ` B ) ) )
64	61, 63	eqtrd 2072	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   =/= wne 2204   E.wrex 2307   _Vcvv 2557   (/)c0 3224   |-> cmpt 3818   Oncon0 4100   suc csuc 4102   omcom 4313   dom cdm 4345   |` cres 4347   Fun wfun 4896   Fn wfn 4897   ` cfv 4902  recscrecs 5919  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecsuc  5991