Theorem frecsuclem3 6013

Description: Lemma for frecsuc 6014. (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 2081	. . . . . . . . . . . . 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 6008	. . . . . . . . . . . . 13 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )
4	1, 3	tfri1d 5972	. . . . . . . . . . . 12 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> recs ( G ) Fn On )
5		fnfun 5016	. . . . . . . . . . . 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 958	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> Fun recs ( G ) )
8		peano2 4336	. . . . . . . . . . 11 |- ( B e. om -> suc B e. om )
9	8	3ad2ant3 961	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> suc B e. om )
10		resfunexg 5403	. . . . . . . . . 10 |- ( ( Fun recs ( G ) /\ suc B e. om ) -> (recs ( G ) |` suc B ) e. _V )
11	7, 9, 10	syl2anc 403	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (recs ( G ) |` suc B ) e. _V )
12		simp1 938	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> A. z ( F ` z ) e. _V )
13		simp2 939	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> A e. V )
14	11, 12, 13	frecabex 6007	. . . . . . . 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 4553	. . . . . . . . . . . . . . . . 17 |- ( g = (recs ( G ) |` suc B ) -> dom g = dom (recs ( G ) |` suc B ) )
16	15	eqeq1d 2089	. . . . . . . . . . . . . . . 16 |- ( g = (recs ( G ) |` suc B ) -> ( dom g = suc m <-> dom (recs ( G ) |` suc B ) = suc m ) )
17		fveq1 5197	. . . . . . . . . . . . . . . . . 18 |- ( g = (recs ( G ) |` suc B ) -> ( g ` m ) = ( (recs ( G ) |` suc B ) ` m ) )
18	17	fveq2d 5202	. . . . . . . . . . . . . . . . 17 |- ( g = (recs ( G ) |` suc B ) -> ( F ` ( g ` m ) ) = ( F ` ( (recs ( G ) |` suc B ) ` m ) ) )
19	18	eleq2d 2148	. . . . . . . . . . . . . . . 16 |- ( g = (recs ( G ) |` suc B ) -> ( x e. ( F ` ( g ` m ) ) <-> x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) ) )
20	16, 19	anbi12d 456	. . . . . . . . . . . . . . 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 2369	. . . . . . . . . . . . . 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 2089	. . . . . . . . . . . . . . 15 |- ( g = (recs ( G ) |` suc B ) -> ( dom g = (/) <-> dom (recs ( G ) |` suc B ) = (/) ) )
23	22	anbi1d 452	. . . . . . . . . . . . . 14 |- ( g = (recs ( G ) |` suc B ) -> ( ( dom g = (/) /\ x e. A ) <-> ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) ) )
24	21, 23	orbi12d 739	. . . . . . . . . . . . 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 2196	. . . . . . . . . . . 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 5269	. . . . . . . . . . 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 113	. . . . . . . . . 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 6010	. . . . . . . . . 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 2089	. . . . . . . . 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 167	. . . . . . . 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 2191	. . . . . 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 6011	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> dom (recs ( G ) |` suc B ) = suc B )
35		peano3 4337	. . . . . . . . . . . 12 |- ( B e. om -> suc B =/= (/) )
36	35	3ad2ant3 961	. . . . . . . . . . 11 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> suc B =/= (/) )
37	34, 36	eqnetrd 2269	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> dom (recs ( G ) |` suc B ) =/= (/) )
38	37	neneqd 2266	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> -. dom (recs ( G ) |` suc B ) = (/) )
39	38	intnanrd 874	. . . . . . . 8 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> -. ( dom (recs ( G ) |` suc B ) = (/) /\ x e. A ) )
40		biorf 695	. . . . . . . 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 679	. . . . . . 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 194	. . . . . 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 2089	. . . . . . . . . 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 2604	. . . . . . . . . . . 12 |- m e. _V
46		suc11g 4300	. . . . . . . . . . . 12 |- ( ( B e. om /\ m e. _V ) -> ( suc B = suc m <-> B = m ) )
47	45, 46	mpan2 415	. . . . . . . . . . 11 |- ( B e. om -> ( suc B = suc m <-> B = m ) )
48	47	3ad2ant3 961	. . . . . . . . . 10 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( suc B = suc m <-> B = m ) )
49	44, 48	bitrd 186	. . . . . . . . 9 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( dom (recs ( G ) |` suc B ) = suc m <-> B = m ) )
50		eqcom 2083	. . . . . . . . 9 |- ( B = m <-> m = B )
51	49, 50	syl6bb 194	. . . . . . . 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 452	. . . . . . 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 2369	. . . . . 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 214	. . . . 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 5198	. . . . . . . 8 |- ( m = B -> ( (recs ( G ) |` suc B ) ` m ) = ( (recs ( G ) |` suc B ) ` B ) )
56	55	fveq2d 5202	. . . . . . 7 |- ( m = B -> ( F ` ( (recs ( G ) |` suc B ) ` m ) ) = ( F ` ( (recs ( G ) |` suc B ) ` B ) ) )
57	56	eleq2d 2148	. . . . . 6 |- ( m = B -> ( x e. ( F ` ( (recs ( G ) |` suc B ) ` m ) ) <-> x e. ( F ` ( (recs ( G ) |` suc B ) ` B ) ) ) )
58	57	ceqsrexbv 2726	. . . . 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 194	. . . 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 1106	. . 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 2079	. 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 6012	. . 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 5202	. 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 2113	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 102   <-> wb 103   \/ wo 661   /\ w3a 919   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   =/= wne 2245   E.wrex 2349   _Vcvv 2601   (/)c0 3251   |-> cmpt 3839   Oncon0 4118   suc csuc 4120   omcom 4331   dom cdm 4363   |` cres 4365   Fun wfun 4916   Fn wfn 4917   ` cfv 4922  recscrecs 5942  freccfrec 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecsuc  6014