Theorem dmfrec 6317

Description: The domain of the finite recursive function generator is the naturals. (Contributed by Scott Fenton, 31-Jul-2019.)

Hypotheses

Ref	Expression
dmfrec.1	|- F = FRec (G, I)
dmfrec.2	|- (ph -> G e. V)
dmfrec.3	|- (ph -> I e. dom G)
dmfrec.4	|- (ph -> ran G C_ dom G)

Assertion

Ref	Expression
dmfrec	|- (ph -> dom F = Nn )

Proof of Theorem dmfrec

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

Step	Hyp	Ref	Expression
1		dmfrec.2	. . . 4 |- (ph -> G e. V)
2		dmfrec.1	. . . . 5 |- F = FRec (G, I)
3	2	frecxpg 6316	. . . 4 |- (G e. V -> F C_ ( Nn X. (ran G u. {I})))
4		dmss 4907	. . . 4 |- (F C_ ( Nn X. (ran G u. {I})) -> dom F C_ dom ( Nn X. (ran G u. {I})))
5	1, 3, 4	3syl 18	. . 3 |- (ph -> dom F C_ dom ( Nn X. (ran G u. {I})))
6		dmxpss 5053	. . 3 |- dom ( Nn X. (ran G u. {I})) C_ Nn
7	5, 6	syl6ss 3285	. 2 |- (ph -> dom F C_ Nn )
8	2	frecexg 6313	. . . . 5 |- (G e. V -> F e. _V)
9	1, 8	syl 15	. . . 4 |- (ph -> F e. _V)
10		dmexg 5106	. . . 4 |- (F e. _V -> dom F e. _V)
11	9, 10	syl 15	. . 3 |- (ph -> dom F e. _V)
12		dmfrec.3	. . . . . . . 8 |- (ph -> I e. dom G)
13		0cex 4393	. . . . . . . . 9 |- 0c e. _V
14		opexg 4588	. . . . . . . . 9 |- ((0c e. _V /\ I e. dom G) -> <.0c, I>. e. _V)
15	13, 14	mpan 651	. . . . . . . 8 |- (I e. dom G -> <.0c, I>. e. _V)
16	12, 15	syl 15	. . . . . . 7 |- (ph -> <.0c, I>. e. _V)
17		snidg 3759	. . . . . . 7 |- (<.0c, I>. e. _V -> <.0c, I>. e. {<.0c, I>.})
18	16, 17	syl 15	. . . . . 6 |- (ph -> <.0c, I>. e. {<.0c, I>.})
19	18	orcd 381	. . . . 5 |- (ph -> (<.0c, I>. e. {<.0c, I>.} \/ E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.0c, I>.))
20		snex 4112	. . . . . 6 |- {<.0c, I>.} e. _V
21		csucex 6260	. . . . . . . 8 |- (w e. _V |-> (w +o 1c)) e. _V
22		pprodexg 5838	. . . . . . . 8 |- (((w e. _V |-> (w +o 1c)) e. _V /\ G e. V) -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
23	21, 22	mpan 651	. . . . . . 7 |- (G e. V -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
24	1, 23	syl 15	. . . . . 6 |- (ph -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
25		df-frec 6311	. . . . . . . 8 |- FRec (G, I) = Clos1 ({<.0c, I>.}, PProd ((w e. _V |-> (w +o 1c)), G))
26	2, 25	eqtri 2373	. . . . . . 7 |- F = Clos1 ({<.0c, I>.}, PProd ((w e. _V |-> (w +o 1c)), G))
27	26	clos1basesucg 5885	. . . . . 6 |- (({<.0c, I>.} e. _V /\ PProd ((w e. _V |-> (w +o 1c)), G) e. _V) -> (<.0c, I>. e. F <-> (<.0c, I>. e. {<.0c, I>.} \/ E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.0c, I>.)))
28	20, 24, 27	sylancr 644	. . . . 5 |- (ph -> (<.0c, I>. e. F <-> (<.0c, I>. e. {<.0c, I>.} \/ E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.0c, I>.)))
29	19, 28	mpbird 223	. . . 4 |- (ph -> <.0c, I>. e. F)
30		opeldm 4911	. . . 4 |- (<.0c, I>. e. F -> 0c e. dom F)
31	29, 30	syl 15	. . 3 |- (ph -> 0c e. dom F)
32		eldm2 4900	. . . . 5 |- (x e. dom F <-> E.y<.x, y>. e. F)
33	26	clos1basesucg 5885	. . . . . . . . . 10 |- (({<.0c, I>.} e. _V /\ PProd ((w e. _V |-> (w +o 1c)), G) e. _V) -> (<.x, y>. e. F <-> (<.x, y>. e. {<.0c, I>.} \/ E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>.)))
34	20, 24, 33	sylancr 644	. . . . . . . . 9 |- (ph -> (<.x, y>. e. F <-> (<.x, y>. e. {<.0c, I>.} \/ E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>.)))
35		vex 2863	. . . . . . . . . . . . . . 15 |- x e. _V
36		vex 2863	. . . . . . . . . . . . . . 15 |- y e. _V
37	35, 36	opex 4589	. . . . . . . . . . . . . 14 |- <.x, y>. e. _V
38	37	elsnc 3757	. . . . . . . . . . . . 13 |- (<.x, y>. e. {<.0c, I>.} <-> <.x, y>. = <.0c, I>.)
39		opth 4603	. . . . . . . . . . . . 13 |- (<.x, y>. = <.0c, I>. <-> (x = 0c /\ y = I))
40	38, 39	bitri 240	. . . . . . . . . . . 12 |- (<.x, y>. e. {<.0c, I>.} <-> (x = 0c /\ y = I))
41	40	simprbi 450	. . . . . . . . . . 11 |- (<.x, y>. e. {<.0c, I>.} -> y = I)
42		eleq1 2413	. . . . . . . . . . . . 13 |- (y = I -> (y e. dom G <-> I e. dom G))
43	42	biimprcd 216	. . . . . . . . . . . 12 |- (I e. dom G -> (y = I -> y e. dom G))
44	12, 43	syl 15	. . . . . . . . . . 11 |- (ph -> (y = I -> y e. dom G))
45	41, 44	syl5 28	. . . . . . . . . 10 |- (ph -> (<.x, y>. e. {<.0c, I>.} -> y e. dom G))
46		opeq 4620	. . . . . . . . . . . . . . . . 17 |- t = <. Proj1 t, Proj2 t>.
47	46	breq1i 4647	. . . . . . . . . . . . . . . 16 |- (t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. <-> <. Proj1 t, Proj2 t>. PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>.)
48		qrpprod 5837	. . . . . . . . . . . . . . . 16 |- (<. Proj1 t, Proj2 t>. PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. <-> ( Proj1 t(w e. _V |-> (w +o 1c))x /\ Proj2 tGy))
49	47, 48	bitri 240	. . . . . . . . . . . . . . 15 |- (t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. <-> ( Proj1 t(w e. _V |-> (w +o 1c))x /\ Proj2 tGy))
50	49	simprbi 450	. . . . . . . . . . . . . 14 |- (t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. -> Proj2 tGy)
51		brelrn 4961	. . . . . . . . . . . . . 14 |- ( Proj2 tGy -> y e. ran G)
52	50, 51	syl 15	. . . . . . . . . . . . 13 |- (t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. -> y e. ran G)
53		dmfrec.4	. . . . . . . . . . . . . 14 |- (ph -> ran G C_ dom G)
54	53	sseld 3273	. . . . . . . . . . . . 13 |- (ph -> (y e. ran G -> y e. dom G))
55	52, 54	syl5 28	. . . . . . . . . . . 12 |- (ph -> (t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. -> y e. dom G))
56	55	adantr 451	. . . . . . . . . . 11 |- ((ph /\ t e. F) -> (t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. -> y e. dom G))
57	56	rexlimdva 2739	. . . . . . . . . 10 |- (ph -> (E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>. -> y e. dom G))
58	45, 57	jaod 369	. . . . . . . . 9 |- (ph -> ((<.x, y>. e. {<.0c, I>.} \/ E.t e. F t PProd ((w e. _V |-> (w +o 1c)), G)<.x, y>.) -> y e. dom G))
59	34, 58	sylbid 206	. . . . . . . 8 |- (ph -> (<.x, y>. e. F -> y e. dom G))
60	59	ancld 536	. . . . . . 7 |- (ph -> (<.x, y>. e. F -> (<.x, y>. e. F /\ y e. dom G)))
61	26	clos1conn 5880	. . . . . . . . 9 |- ((<.x, y>. e. F /\ <.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.) -> <.(x +o 1c), z>. e. F)
62	61	eximi 1576	. . . . . . . 8 |- (E.z(<.x, y>. e. F /\ <.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.) -> E.z<.(x +o 1c), z>. e. F)
63		eldm 4899	. . . . . . . . . . 11 |- (y e. dom G <-> E.z yGz)
64		eqid 2353	. . . . . . . . . . . . . 14 |- (x +o 1c) = (x +o 1c)
65		1cex 4143	. . . . . . . . . . . . . . . 16 |- 1c e. _V
66	35, 65	addcex 4395	. . . . . . . . . . . . . . 15 |- (x +o 1c) e. _V
67	35, 66	brcsuc 6261	. . . . . . . . . . . . . 14 |- (x(w e. _V |-> (w +o 1c))(x +o 1c) <-> (x +o 1c) = (x +o 1c))
68	64, 67	mpbir 200	. . . . . . . . . . . . 13 |- x(w e. _V |-> (w +o 1c))(x +o 1c)
69		qrpprod 5837	. . . . . . . . . . . . 13 |- (<.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>. <-> (x(w e. _V |-> (w +o 1c))(x +o 1c) /\ yGz))
70	68, 69	mpbiran 884	. . . . . . . . . . . 12 |- (<.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>. <-> yGz)
71	70	exbii 1582	. . . . . . . . . . 11 |- (E.z<.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>. <-> E.z yGz)
72	63, 71	bitr4i 243	. . . . . . . . . 10 |- (y e. dom G <-> E.z<.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.)
73	72	anbi2i 675	. . . . . . . . 9 |- ((<.x, y>. e. F /\ y e. dom G) <-> (<.x, y>. e. F /\ E.z<.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.))
74		19.42v 1905	. . . . . . . . 9 |- (E.z(<.x, y>. e. F /\ <.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.) <-> (<.x, y>. e. F /\ E.z<.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.))
75	73, 74	bitr4i 243	. . . . . . . 8 |- ((<.x, y>. e. F /\ y e. dom G) <-> E.z(<.x, y>. e. F /\ <.x, y>. PProd ((w e. _V |-> (w +o 1c)), G)<.(x +o 1c), z>.))
76		eldm2 4900	. . . . . . . 8 |- ((x +o 1c) e. dom F <-> E.z<.(x +o 1c), z>. e. F)
77	62, 75, 76	3imtr4i 257	. . . . . . 7 |- ((<.x, y>. e. F /\ y e. dom G) -> (x +o 1c) e. dom F)
78	60, 77	syl6 29	. . . . . 6 |- (ph -> (<.x, y>. e. F -> (x +o 1c) e. dom F))
79	78	exlimdv 1636	. . . . 5 |- (ph -> (E.y<.x, y>. e. F -> (x +o 1c) e. dom F))
80	32, 79	syl5bi 208	. . . 4 |- (ph -> (x e. dom F -> (x +o 1c) e. dom F))
81	80	ralrimivw 2699	. . 3 |- (ph -> A.x e. Nn (x e. dom F -> (x +o 1c) e. dom F))
82		peano5 4410	. . 3 |- ((dom F e. _V /\ 0c e. dom F /\ A.x e. Nn (x e. dom F -> (x +o 1c) e. dom F)) -> Nn C_ dom F)
83	11, 31, 81, 82	syl3anc 1182	. 2 |- (ph -> Nn C_ dom F)
84	7, 83	eqssd 3290	1 |- (ph -> dom F = Nn )

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  A.wral 2615  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   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640   X. cxp 4771  dom cdm 4773  ran crn 4774   |-> cmpt 5652   PProd cpprod 5738   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-clos1 5874  df-frec 6311

This theorem is referenced by:  fnfreclem3  6320  fnfrec  6321  frecsuc  6323