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Theorem frec0g 6668

Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)

**Assertion**
Ref	Expression
frec0g	frec

Proof of Theorem frec0g Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dm0 4995	. . . . . . . . . 10
2	1	biantrur 303	. . . . . . . . 9 $/\$
3		vex 2824	. . . . . . . . . . . . . . . 16
4		nsuceq0g 4563	. . . . . . . . . . . . . . . 16
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . 15
6	5	nesymi 2466	. . . . . . . . . . . . . 14
7	1	eqeq1i 2246	. . . . . . . . . . . . . 14
8	6, 7	mtbir 682	. . . . . . . . . . . . 13
9	8	intnanr 942	. . . . . . . . . . . 12 $/\$
10	9	a1i 9	. . . . . . . . . . 11 $/\$
11	10	nrex 2642	. . . . . . . . . 10 $/\$
12	11	biorfi 758	. . . . . . . . 9 $/\$ $/\$ $\/$ $/\$
13		orcom 740	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
14	2, 12, 13	3bitri 206	. . . . . . . 8 $/\$ $\/$ $/\$
15	14	abbii 2354	. . . . . . 7 $/\$ $\/$ $/\$
16		abid2 2361	. . . . . . 7
17	15, 16	eqtr3i 2261	. . . . . 6 $/\$ $\/$ $/\$
18		elex 2833	. . . . . 6
19	17, 18	eqeltrid 2325	. . . . 5 $/\$ $\/$ $/\$
20		0ex 4260	. . . . . . 7
21		dmeq 4981	. . . . . . . . . . . . 13
22	21	eqeq1d 2247	. . . . . . . . . . . 12
23		fveq1 5694	. . . . . . . . . . . . . 14
24	23	fveq2d 5699	. . . . . . . . . . . . 13
25	24	eleq2d 2308	. . . . . . . . . . . 12
26	22, 25	anbi12d 477	. . . . . . . . . . 11 $/\$ $/\$
27	26	rexbidv 2551	. . . . . . . . . 10 $/\$ $/\$
28	21	eqeq1d 2247	. . . . . . . . . . 11
29	28	anbi1d 469	. . . . . . . . . 10 $/\$ $/\$
30	27, 29	orbi12d 805	. . . . . . . . 9 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
31	30	abbidv 2358	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
32		eqid 2238	. . . . . . . 8 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
33	31, 32	fvmptg 5781	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
34	20, 33	mpan 428	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
35	34, 17	eqtrdi 2287	. . . . 5 $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
36	19, 35	syl 14	. . . 4 $/\$ $\/$ $/\$
37	36, 18	eqeltrd 2315	. . 3 $/\$ $\/$ $/\$
38		df-frec 6662	. . . . . 6 frec recs $/\$ $\/$ $/\$
39	38	fveq1i 5696	. . . . 5 frec recs $/\$ $\/$ $/\$
40		peano1 4741	. . . . . 6
41		fvres 5719	. . . . . 6 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
42	40, 41	ax-mp 5	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
43	39, 42	eqtri 2259	. . . 4 frec recs $/\$ $\/$ $/\$
44		eqid 2238	. . . . 5 recs $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$
45	44	tfr0 6594	. . . 4 $/\$ $\/$ $/\$ recs $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
46	43, 45	eqtrid 2283	. . 3 $/\$ $\/$ $/\$ frec $/\$ $\/$ $/\$
47	37, 46	syl 14	. 2 frec $/\$ $\/$ $/\$
48	47, 36	eqtrd 2271	1 frec

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104 $\/$ wo 720

wceq 1402

wcel 2209

cab 2224

wne 2420

wrex 2529

cvv 2821

c0 3520

cmpt 4192

csuc 4510

com 4737

cdm 4774

cres 4776

cfv 5377 recscrecs 6575 freccfrec 6661

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-res 4786 df-iota 5337 df-fun 5379 df-fn 5380 df-fv 5385 df-recs 6576 df-frec 6662

This theorem is used by: frecrdg 6679 frec2uz0d 10836 frec2uzrdg 10846 frecuzrdg0 10850 frecuzrdgg 10853 frecuzrdg0t 10859 seq3val 10897 seqvalcd 10898

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