Theorem frecsuc 6323

Description: Calculate the value of the finite recursive function generator at a successor. (Contributed by Scott Fenton, 31-Jul-2019.)

Hypotheses

Ref	Expression
frecsuc.1	|- F = FRec (G, I)
frecsuc.2	|- (ph -> G e. Funs )
frecsuc.3	|- (ph -> I e. dom G)
frecsuc.4	|- (ph -> ran G C_ dom G)
frecsuc.5	|- (ph -> X e. Nn )

Assertion

Ref	Expression
frecsuc	|- (ph -> (F` (X +o 1c)) = (G` (F` X)))

Proof of Theorem frecsuc

Dummy variables w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecsuc.1	. . . . . . . 8 |- F = FRec (G, I)
2		frecsuc.2	. . . . . . . 8 |- (ph -> G e. Funs )
3		frecsuc.3	. . . . . . . 8 |- (ph -> I e. dom G)
4		frecsuc.4	. . . . . . . 8 |- (ph -> ran G C_ dom G)
5	1, 2, 3, 4	fnfrec 6321	. . . . . . 7 |- (ph -> F Fn Nn )
6		fnfun 5182	. . . . . . 7 |- (F Fn Nn -> Fun F)
7	5, 6	syl 15	. . . . . 6 |- (ph -> Fun F)
8		frecsuc.5	. . . . . . 7 |- (ph -> X e. Nn )
9	1, 2, 3, 4	dmfrec 6317	. . . . . . 7 |- (ph -> dom F = Nn )
10	8, 9	eleqtrrd 2430	. . . . . 6 |- (ph -> X e. dom F)
11		funfvop 5401	. . . . . 6 |- ((Fun F /\ X e. dom F) -> <.X, (F` X)>. e. F)
12	7, 10, 11	syl2anc 642	. . . . 5 |- (ph -> <.X, (F` X)>. e. F)
13		eqid 2353	. . . . . 6 |- (X +o 1c) = (X +o 1c)
14		peano2 4404	. . . . . . . 8 |- (X e. Nn -> (X +o 1c) e. Nn )
15	8, 14	syl 15	. . . . . . 7 |- (ph -> (X +o 1c) e. Nn )
16		addceq1 4384	. . . . . . . . 9 |- (w = X -> (w +o 1c) = (X +o 1c))
17	16	eqeq2d 2364	. . . . . . . 8 |- (w = X -> (y = (w +o 1c) <-> y = (X +o 1c)))
18		eqeq1 2359	. . . . . . . 8 |- (y = (X +o 1c) -> (y = (X +o 1c) <-> (X +o 1c) = (X +o 1c)))
19		mptv 5719	. . . . . . . 8 |- (w e. _V |-> (w +o 1c)) = {<.w, y>. | y = (w +o 1c)}
20	17, 18, 19	brabg 4707	. . . . . . 7 |- ((X e. Nn /\ (X +o 1c) e. Nn ) -> (X(w e. _V |-> (w +o 1c))(X +o 1c) <-> (X +o 1c) = (X +o 1c)))
21	8, 15, 20	syl2anc 642	. . . . . 6 |- (ph -> (X(w e. _V |-> (w +o 1c))(X +o 1c) <-> (X +o 1c) = (X +o 1c)))
22	13, 21	mpbiri 224	. . . . 5 |- (ph -> X(w e. _V |-> (w +o 1c))(X +o 1c))
23		elfunsi 5832	. . . . . . . 8 |- (G e. Funs -> Fun G)
24	2, 23	syl 15	. . . . . . 7 |- (ph -> Fun G)
25	3	snssd 3854	. . . . . . . . 9 |- (ph -> {I} C_ dom G)
26	4, 25	unssd 3440	. . . . . . . 8 |- (ph -> (ran G u. {I}) C_ dom G)
27	1	frecxpg 6316	. . . . . . . . . . . 12 |- (G e. Funs -> F C_ ( Nn X. (ran G u. {I})))
28	2, 27	syl 15	. . . . . . . . . . 11 |- (ph -> F C_ ( Nn X. (ran G u. {I})))
29		rnss 4960	. . . . . . . . . . 11 |- (F C_ ( Nn X. (ran G u. {I})) -> ran F C_ ran ( Nn X. (ran G u. {I})))
30	28, 29	syl 15	. . . . . . . . . 10 |- (ph -> ran F C_ ran ( Nn X. (ran G u. {I})))
31		rnxpss 5054	. . . . . . . . . 10 |- ran ( Nn X. (ran G u. {I})) C_ (ran G u. {I})
32	30, 31	syl6ss 3285	. . . . . . . . 9 |- (ph -> ran F C_ (ran G u. {I}))
33		fvelrn 5414	. . . . . . . . . 10 |- ((Fun F /\ X e. dom F) -> (F` X) e. ran F)
34	7, 10, 33	syl2anc 642	. . . . . . . . 9 |- (ph -> (F` X) e. ran F)
35	32, 34	sseldd 3275	. . . . . . . 8 |- (ph -> (F` X) e. (ran G u. {I}))
36	26, 35	sseldd 3275	. . . . . . 7 |- (ph -> (F` X) e. dom G)
37		funfvop 5401	. . . . . . 7 |- ((Fun G /\ (F` X) e. dom G) -> <.(F` X), (G` (F` X))>. e. G)
38	24, 36, 37	syl2anc 642	. . . . . 6 |- (ph -> <.(F` X), (G` (F` X))>. e. G)
39		df-br 4641	. . . . . 6 |- ((F` X)G(G` (F` X)) <-> <.(F` X), (G` (F` X))>. e. G)
40	38, 39	sylibr 203	. . . . 5 |- (ph -> (F` X)G(G` (F` X)))
41		breq1 4643	. . . . . . 7 |- (y = <.X, (F` X)>. -> (y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>. <-> <.X, (F` X)>. PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>.))
42		qrpprod 5837	. . . . . . 7 |- (<.X, (F` X)>. PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>. <-> (X(w e. _V |-> (w +o 1c))(X +o 1c) /\ (F` X)G(G` (F` X))))
43	41, 42	syl6bb 252	. . . . . 6 |- (y = <.X, (F` X)>. -> (y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>. <-> (X(w e. _V |-> (w +o 1c))(X +o 1c) /\ (F` X)G(G` (F` X)))))
44	43	rspcev 2956	. . . . 5 |- ((<.X, (F` X)>. e. F /\ (X(w e. _V |-> (w +o 1c))(X +o 1c) /\ (F` X)G(G` (F` X)))) -> E.y e. F y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>.)
45	12, 22, 40, 44	syl12anc 1180	. . . 4 |- (ph -> E.y e. F y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>.)
46	45	olcd 382	. . 3 |- (ph -> (<.(X +o 1c), (G` (F` X))>. e. {<.0c, I>.} \/ E.y e. F y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>.))
47		snex 4112	. . . 4 |- {<.0c, I>.} e. _V
48		csucex 6260	. . . . 5 |- (w e. _V |-> (w +o 1c)) e. _V
49		pprodexg 5838	. . . . 5 |- (((w e. _V |-> (w +o 1c)) e. _V /\ G e. Funs ) -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
50	48, 2, 49	sylancr 644	. . . 4 |- (ph -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
51		df-frec 6311	. . . . . 6 |- FRec (G, I) = Clos1 ({<.0c, I>.}, PProd ((w e. _V |-> (w +o 1c)), G))
52	1, 51	eqtri 2373	. . . . 5 |- F = Clos1 ({<.0c, I>.}, PProd ((w e. _V |-> (w +o 1c)), G))
53	52	clos1basesucg 5885	. . . 4 |- (({<.0c, I>.} e. _V /\ PProd ((w e. _V |-> (w +o 1c)), G) e. _V) -> (<.(X +o 1c), (G` (F` X))>. e. F <-> (<.(X +o 1c), (G` (F` X))>. e. {<.0c, I>.} \/ E.y e. F y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>.)))
54	47, 50, 53	sylancr 644	. . 3 |- (ph -> (<.(X +o 1c), (G` (F` X))>. e. F <-> (<.(X +o 1c), (G` (F` X))>. e. {<.0c, I>.} \/ E.y e. F y PProd ((w e. _V |-> (w +o 1c)), G)<.(X +o 1c), (G` (F` X))>.)))
55	46, 54	mpbird 223	. 2 |- (ph -> <.(X +o 1c), (G` (F` X))>. e. F)
56		fnopfvb 5360	. . 3 |- ((F Fn Nn /\ (X +o 1c) e. Nn ) -> ((F` (X +o 1c)) = (G` (F` X)) <-> <.(X +o 1c), (G` (F` X))>. e. F))
57	5, 15, 56	syl2anc 642	. 2 |- (ph -> ((F` (X +o 1c)) = (G` (F` X)) <-> <.(X +o 1c), (G` (F` X))>. e. F))
58	55, 57	mpbird 223	1 |- (ph -> (F` (X +o 1c)) = (G` (F` X)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860   u. cun 3208   C_ wss 3258  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376  <.cop 4562   class class class wbr 4640   X. cxp 4771  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  ` cfv 4782   |-> cmpt 5652   PProd cpprod 5738   Funs cfuns 5760   Clos1 cclos1 5873   FRec cfrec 6310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-pprod 5739  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-clos1 5874  df-frec 6311

This theorem is referenced by: (None)