Theorem fnfreclem2 6319

Description: Lemma for fnfrec 6321. Calculate the unique value of F at zero. (Contributed by Scott Fenton, 31-Jul-2019.)

Hypotheses

Ref	Expression
fnfreclem2.1	|- F = FRec (G, I)
fnfreclem2.2	|- (ph -> G e. V)
fnfreclem2.3	|- (ph -> I e. dom G)
fnfreclem2.4	|- (ph -> ran G C_ dom G)

Assertion

Ref	Expression
fnfreclem2	|- (ph -> (0cFX -> X = I))

Proof of Theorem fnfreclem2

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

Step	Hyp	Ref	Expression
1		df-br 4641	. 2 |- (0cFX <-> <.0c, X>. e. F)
2		snex 4112	. . . 4 |- {<.0c, I>.} e. _V
3		csucex 6260	. . . . 5 |- (w e. _V |-> (w +o 1c)) e. _V
4		fnfreclem2.2	. . . . 5 |- (ph -> G e. V)
5		pprodexg 5838	. . . . 5 |- (((w e. _V |-> (w +o 1c)) e. _V /\ G e. V) -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
6	3, 4, 5	sylancr 644	. . . 4 |- (ph -> PProd ((w e. _V |-> (w +o 1c)), G) e. _V)
7		fnfreclem2.1	. . . . . 6 |- F = FRec (G, I)
8		df-frec 6311	. . . . . 6 |- FRec (G, I) = Clos1 ({<.0c, I>.}, PProd ((w e. _V |-> (w +o 1c)), G))
9	7, 8	eqtri 2373	. . . . 5 |- F = Clos1 ({<.0c, I>.}, PProd ((w e. _V |-> (w +o 1c)), G))
10	9	clos1basesucg 5885	. . . 4 |- (({<.0c, I>.} e. _V /\ PProd ((w e. _V |-> (w +o 1c)), G) e. _V) -> (<.0c, X>. e. F <-> (<.0c, X>. e. {<.0c, I>.} \/ E.z e. F z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.)))
11	2, 6, 10	sylancr 644	. . 3 |- (ph -> (<.0c, X>. e. F <-> (<.0c, X>. e. {<.0c, I>.} \/ E.z e. F z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.)))
12		0cex 4393	. . . . . . 7 |- 0c e. _V
13		fnfreclem2.3	. . . . . . 7 |- (ph -> I e. dom G)
14		opexg 4588	. . . . . . 7 |- ((0c e. _V /\ I e. dom G) -> <.0c, I>. e. _V)
15	12, 13, 14	sylancr 644	. . . . . 6 |- (ph -> <.0c, I>. e. _V)
16		elsnc2g 3762	. . . . . 6 |- (<.0c, I>. e. _V -> (<.0c, X>. e. {<.0c, I>.} <-> <.0c, X>. = <.0c, I>.))
17	15, 16	syl 15	. . . . 5 |- (ph -> (<.0c, X>. e. {<.0c, I>.} <-> <.0c, X>. = <.0c, I>.))
18		opth 4603	. . . . . 6 |- (<.0c, X>. = <.0c, I>. <-> (0c = 0c /\ X = I))
19	18	simprbi 450	. . . . 5 |- (<.0c, X>. = <.0c, I>. -> X = I)
20	17, 19	syl6bi 219	. . . 4 |- (ph -> (<.0c, X>. e. {<.0c, I>.} -> X = I))
21		0cnsuc 4402	. . . . . . . . . . 11 |- ( Proj1 z +o 1c) =/= 0c
22		df-ne 2519	. . . . . . . . . . 11 |- (( Proj1 z +o 1c) =/= 0c <-> -. ( Proj1 z +o 1c) = 0c)
23	21, 22	mpbi 199	. . . . . . . . . 10 |- -. ( Proj1 z +o 1c) = 0c
24	23	intnanr 881	. . . . . . . . 9 |- -. (( Proj1 z +o 1c) = 0c /\ Proj2 zGX)
25		qrpprod 5837	. . . . . . . . . 10 |- (<. Proj1 z, Proj2 z>. PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>. <-> ( Proj1 z(w e. _V |-> (w +o 1c))0c /\ Proj2 zGX))
26		opeq 4620	. . . . . . . . . . 11 |- z = <. Proj1 z, Proj2 z>.
27	26	breq1i 4647	. . . . . . . . . 10 |- (z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>. <-> <. Proj1 z, Proj2 z>. PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.)
28		vex 2863	. . . . . . . . . . . . . . 15 |- z e. _V
29	28	proj1ex 4594	. . . . . . . . . . . . . 14 |- Proj1 z e. _V
30		addceq1 4384	. . . . . . . . . . . . . . 15 |- (w = Proj1 z -> (w +o 1c) = ( Proj1 z +o 1c))
31		eqid 2353	. . . . . . . . . . . . . . 15 |- (w e. _V |-> (w +o 1c)) = (w e. _V |-> (w +o 1c))
32		1cex 4143	. . . . . . . . . . . . . . . 16 |- 1c e. _V
33	29, 32	addcex 4395	. . . . . . . . . . . . . . 15 |- ( Proj1 z +o 1c) e. _V
34	30, 31, 33	fvmpt 5701	. . . . . . . . . . . . . 14 |- ( Proj1 z e. _V -> ((w e. _V |-> (w +o 1c))` Proj1 z) = ( Proj1 z +o 1c))
35	29, 34	ax-mp 5	. . . . . . . . . . . . 13 |- ((w e. _V |-> (w +o 1c))` Proj1 z) = ( Proj1 z +o 1c)
36	35	eqeq1i 2360	. . . . . . . . . . . 12 |- (((w e. _V |-> (w +o 1c))` Proj1 z) = 0c <-> ( Proj1 z +o 1c) = 0c)
37		vex 2863	. . . . . . . . . . . . . . 15 |- w e. _V
38	37, 32	addcex 4395	. . . . . . . . . . . . . 14 |- (w +o 1c) e. _V
39	38, 31	fnmpti 5691	. . . . . . . . . . . . 13 |- (w e. _V |-> (w +o 1c)) Fn _V
40		fnbrfvb 5359	. . . . . . . . . . . . 13 |- (((w e. _V |-> (w +o 1c)) Fn _V /\ Proj1 z e. _V) -> (((w e. _V |-> (w +o 1c))` Proj1 z) = 0c <-> Proj1 z(w e. _V |-> (w +o 1c))0c))
41	39, 29, 40	mp2an 653	. . . . . . . . . . . 12 |- (((w e. _V |-> (w +o 1c))` Proj1 z) = 0c <-> Proj1 z(w e. _V |-> (w +o 1c))0c)
42	36, 41	bitr3i 242	. . . . . . . . . . 11 |- (( Proj1 z +o 1c) = 0c <-> Proj1 z(w e. _V |-> (w +o 1c))0c)
43	42	anbi1i 676	. . . . . . . . . 10 |- ((( Proj1 z +o 1c) = 0c /\ Proj2 zGX) <-> ( Proj1 z(w e. _V |-> (w +o 1c))0c /\ Proj2 zGX))
44	25, 27, 43	3bitr4i 268	. . . . . . . . 9 |- (z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>. <-> (( Proj1 z +o 1c) = 0c /\ Proj2 zGX))
45	24, 44	mtbir 290	. . . . . . . 8 |- -. z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.
46	45	a1i 10	. . . . . . 7 |- (z e. F -> -. z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.)
47	46	nrex 2717	. . . . . 6 |- -. E.z e. F z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.
48	47	pm2.21i 123	. . . . 5 |- (E.z e. F z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>. -> X = I)
49	48	a1i 10	. . . 4 |- (ph -> (E.z e. F z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>. -> X = I))
50	20, 49	jaod 369	. . 3 |- (ph -> ((<.0c, X>. e. {<.0c, I>.} \/ E.z e. F z PProd ((w e. _V |-> (w +o 1c)), G)<.0c, X>.) -> X = I))
51	11, 50	sylbid 206	. 2 |- (ph -> (<.0c, X>. e. F -> X = I))
52	1, 51	syl5bi 208	1 |- (ph -> (0cFX -> X = I))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  _Vcvv 2860   C_ wss 3258  {csn 3738  1_cc1c 4135  0_cc0c 4375   +o cplc 4376  <.cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640  dom cdm 4773  ran crn 4774   Fn wfn 4777  ` cfv 4782   |-> 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:  fnfrec  6321