Theorem fnfreclem3 6317

Description: Lemma for fnfrec 6318. The value of F at a successor is G related to a previous element. (Contributed by Scott Fenton, 31-Jul-2019.)

Hypotheses

Ref	Expression
fnfreclem2.1	⊢ F = FRec (G, I)
fnfreclem2.2	⊢ (φ → G ∈ V)
fnfreclem2.3	⊢ (φ → I ∈ dom G)
fnfreclem2.4	⊢ (φ → ran G ⊆ dom G)
fnfreclem3.5	⊢ (φ → X ∈ Nn )
fnfreclem3.6	⊢ (φ → (X +c 1c)FY)

Assertion

Ref	Expression
fnfreclem3	⊢ (φ → ∃z(XFz ∧ zGY))

Distinct variable groups:   z,G   z,I   z,X   φ,z   z,F   z,Y

Allowed substitution hint:   V(z)

Proof of Theorem fnfreclem3

Dummy variables w a t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cex 4392	. . . . 5 ⊢ 0c ∈ V
2		fnfreclem2.3	. . . . 5 ⊢ (φ → I ∈ dom G)
3		opexg 4587	. . . . 5 ⊢ ((0c ∈ V ∧ I ∈ dom G) → ⟨0c, I⟩ ∈ V)
4	1, 2, 3	sylancr 644	. . . 4 ⊢ (φ → ⟨0c, I⟩ ∈ V)
5		elsnc2g 3761	. . . 4 ⊢ (⟨0c, I⟩ ∈ V → (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} ↔ ⟨(X +c 1c), Y⟩ = ⟨0c, I⟩))
6	4, 5	syl 15	. . 3 ⊢ (φ → (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} ↔ ⟨(X +c 1c), Y⟩ = ⟨0c, I⟩))
7		opth 4602	. . . . 5 ⊢ (⟨(X +c 1c), Y⟩ = ⟨0c, I⟩ ↔ ((X +c 1c) = 0c ∧ Y = I))
8	7	simplbi 446	. . . 4 ⊢ (⟨(X +c 1c), Y⟩ = ⟨0c, I⟩ → (X +c 1c) = 0c)
9		0cnsuc 4401	. . . . . . 7 ⊢ (X +c 1c) ≠ 0c
10		df-ne 2518	. . . . . . 7 ⊢ ((X +c 1c) ≠ 0c ↔ ¬ (X +c 1c) = 0c)
11	9, 10	mpbi 199	. . . . . 6 ⊢ ¬ (X +c 1c) = 0c
12	11	pm2.21i 123	. . . . 5 ⊢ ((X +c 1c) = 0c → ∃z(XFz ∧ zGY))
13	12	a1i 10	. . . 4 ⊢ (φ → ((X +c 1c) = 0c → ∃z(XFz ∧ zGY)))
14	8, 13	syl5 28	. . 3 ⊢ (φ → (⟨(X +c 1c), Y⟩ = ⟨0c, I⟩ → ∃z(XFz ∧ zGY)))
15	6, 14	sylbid 206	. 2 ⊢ (φ → (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} → ∃z(XFz ∧ zGY)))
16		vex 2862	. . . . . . 7 ⊢ a ∈ V
17		opeqex 4621	. . . . . . 7 ⊢ (a ∈ V → ∃t∃z a = ⟨t, z⟩)
18	16, 17	ax-mp 8	. . . . . 6 ⊢ ∃t∃z a = ⟨t, z⟩
19		excom 1741	. . . . . 6 ⊢ (∃t∃z a = ⟨t, z⟩ ↔ ∃z∃t a = ⟨t, z⟩)
20	18, 19	mpbi 199	. . . . 5 ⊢ ∃z∃t a = ⟨t, z⟩
21		eleq1 2413	. . . . . . . . . . . 12 ⊢ (a = ⟨t, z⟩ → (a ∈ F ↔ ⟨t, z⟩ ∈ F))
22		df-br 4640	. . . . . . . . . . . 12 ⊢ (tFz ↔ ⟨t, z⟩ ∈ F)
23	21, 22	syl6bbr 254	. . . . . . . . . . 11 ⊢ (a = ⟨t, z⟩ → (a ∈ F ↔ tFz))
24	23	anbi2d 684	. . . . . . . . . 10 ⊢ (a = ⟨t, z⟩ → ((φ ∧ a ∈ F) ↔ (φ ∧ tFz)))
25		breq1 4642	. . . . . . . . . . 11 ⊢ (a = ⟨t, z⟩ → (a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩ ↔ ⟨t, z⟩ PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩))
26		qrpprod 5836	. . . . . . . . . . . 12 ⊢ (⟨t, z⟩ PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩ ↔ (t(w ∈ V ↦ (w +c 1c))(X +c 1c) ∧ zGY))
27		vex 2862	. . . . . . . . . . . . . . . 16 ⊢ t ∈ V
28		addceq1 4383	. . . . . . . . . . . . . . . . 17 ⊢ (w = t → (w +c 1c) = (t +c 1c))
29		eqid 2353	. . . . . . . . . . . . . . . . 17 ⊢ (w ∈ V ↦ (w +c 1c)) = (w ∈ V ↦ (w +c 1c))
30		1cex 4142	. . . . . . . . . . . . . . . . . 18 ⊢ 1c ∈ V
31	27, 30	addcex 4394	. . . . . . . . . . . . . . . . 17 ⊢ (t +c 1c) ∈ V
32	28, 29, 31	fvmpt 5700	. . . . . . . . . . . . . . . 16 ⊢ (t ∈ V → ((w ∈ V ↦ (w +c 1c)) ‘t) = (t +c 1c))
33	27, 32	ax-mp 8	. . . . . . . . . . . . . . 15 ⊢ ((w ∈ V ↦ (w +c 1c)) ‘t) = (t +c 1c)
34	33	eqeq1i 2360	. . . . . . . . . . . . . 14 ⊢ (((w ∈ V ↦ (w +c 1c)) ‘t) = (X +c 1c) ↔ (t +c 1c) = (X +c 1c))
35	29	fnmpt 5689	. . . . . . . . . . . . . . . 16 ⊢ (∀w ∈ V (w +c 1c) ∈ V → (w ∈ V ↦ (w +c 1c)) Fn V)
36		addcexg 4393	. . . . . . . . . . . . . . . . 17 ⊢ ((w ∈ V ∧ 1c ∈ V) → (w +c 1c) ∈ V)
37	30, 36	mpan2 652	. . . . . . . . . . . . . . . 16 ⊢ (w ∈ V → (w +c 1c) ∈ V)
38	35, 37	mprg 2683	. . . . . . . . . . . . . . 15 ⊢ (w ∈ V ↦ (w +c 1c)) Fn V
39		fnbrfvb 5358	. . . . . . . . . . . . . . 15 ⊢ (((w ∈ V ↦ (w +c 1c)) Fn V ∧ t ∈ V) → (((w ∈ V ↦ (w +c 1c)) ‘t) = (X +c 1c) ↔ t(w ∈ V ↦ (w +c 1c))(X +c 1c)))
40	38, 27, 39	mp2an 653	. . . . . . . . . . . . . 14 ⊢ (((w ∈ V ↦ (w +c 1c)) ‘t) = (X +c 1c) ↔ t(w ∈ V ↦ (w +c 1c))(X +c 1c))
41	34, 40	bitr3i 242	. . . . . . . . . . . . 13 ⊢ ((t +c 1c) = (X +c 1c) ↔ t(w ∈ V ↦ (w +c 1c))(X +c 1c))
42	41	anbi1i 676	. . . . . . . . . . . 12 ⊢ (((t +c 1c) = (X +c 1c) ∧ zGY) ↔ (t(w ∈ V ↦ (w +c 1c))(X +c 1c) ∧ zGY))
43	26, 42	bitr4i 243	. . . . . . . . . . 11 ⊢ (⟨t, z⟩ PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩ ↔ ((t +c 1c) = (X +c 1c) ∧ zGY))
44	25, 43	syl6bb 252	. . . . . . . . . 10 ⊢ (a = ⟨t, z⟩ → (a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩ ↔ ((t +c 1c) = (X +c 1c) ∧ zGY)))
45	24, 44	anbi12d 691	. . . . . . . . 9 ⊢ (a = ⟨t, z⟩ → (((φ ∧ a ∈ F) ∧ a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩) ↔ ((φ ∧ tFz) ∧ ((t +c 1c) = (X +c 1c) ∧ zGY))))
46		breldm 4911	. . . . . . . . . . . . . . 15 ⊢ (tFz → t ∈ dom F)
47	46	adantl 452	. . . . . . . . . . . . . 14 ⊢ ((φ ∧ tFz) → t ∈ dom F)
48		fnfreclem2.1	. . . . . . . . . . . . . . . 16 ⊢ F = FRec (G, I)
49		fnfreclem2.2	. . . . . . . . . . . . . . . 16 ⊢ (φ → G ∈ V)
50		fnfreclem2.4	. . . . . . . . . . . . . . . 16 ⊢ (φ → ran G ⊆ dom G)
51	48, 49, 2, 50	dmfrec 6314	. . . . . . . . . . . . . . 15 ⊢ (φ → dom F = Nn )
52	51	adantr 451	. . . . . . . . . . . . . 14 ⊢ ((φ ∧ tFz) → dom F = Nn )
53	47, 52	eleqtrd 2429	. . . . . . . . . . . . 13 ⊢ ((φ ∧ tFz) → t ∈ Nn )
54		fnfreclem3.5	. . . . . . . . . . . . . 14 ⊢ (φ → X ∈ Nn )
55	54	adantr 451	. . . . . . . . . . . . 13 ⊢ ((φ ∧ tFz) → X ∈ Nn )
56		peano4 4557	. . . . . . . . . . . . . 14 ⊢ ((t ∈ Nn ∧ X ∈ Nn ∧ (t +c 1c) = (X +c 1c)) → t = X)
57	56	3expia 1153	. . . . . . . . . . . . 13 ⊢ ((t ∈ Nn ∧ X ∈ Nn ) → ((t +c 1c) = (X +c 1c) → t = X))
58	53, 55, 57	syl2anc 642	. . . . . . . . . . . 12 ⊢ ((φ ∧ tFz) → ((t +c 1c) = (X +c 1c) → t = X))
59		breq1 4642	. . . . . . . . . . . . . 14 ⊢ (t = X → (tFz ↔ XFz))
60	59	biimpcd 215	. . . . . . . . . . . . 13 ⊢ (tFz → (t = X → XFz))
61	60	adantl 452	. . . . . . . . . . . 12 ⊢ ((φ ∧ tFz) → (t = X → XFz))
62	58, 61	syld 40	. . . . . . . . . . 11 ⊢ ((φ ∧ tFz) → ((t +c 1c) = (X +c 1c) → XFz))
63	62	anim1d 547	. . . . . . . . . 10 ⊢ ((φ ∧ tFz) → (((t +c 1c) = (X +c 1c) ∧ zGY) → (XFz ∧ zGY)))
64	63	imp 418	. . . . . . . . 9 ⊢ (((φ ∧ tFz) ∧ ((t +c 1c) = (X +c 1c) ∧ zGY)) → (XFz ∧ zGY))
65	45, 64	syl6bi 219	. . . . . . . 8 ⊢ (a = ⟨t, z⟩ → (((φ ∧ a ∈ F) ∧ a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩) → (XFz ∧ zGY)))
66	65	com12 27	. . . . . . 7 ⊢ (((φ ∧ a ∈ F) ∧ a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩) → (a = ⟨t, z⟩ → (XFz ∧ zGY)))
67	66	exlimdv 1636	. . . . . 6 ⊢ (((φ ∧ a ∈ F) ∧ a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩) → (∃t a = ⟨t, z⟩ → (XFz ∧ zGY)))
68	67	eximdv 1622	. . . . 5 ⊢ (((φ ∧ a ∈ F) ∧ a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩) → (∃z∃t a = ⟨t, z⟩ → ∃z(XFz ∧ zGY)))
69	20, 68	mpi 16	. . . 4 ⊢ (((φ ∧ a ∈ F) ∧ a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩) → ∃z(XFz ∧ zGY))
70	69	ex 423	. . 3 ⊢ ((φ ∧ a ∈ F) → (a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩ → ∃z(XFz ∧ zGY)))
71	70	rexlimdva 2738	. 2 ⊢ (φ → (∃a ∈ F a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩ → ∃z(XFz ∧ zGY)))
72		fnfreclem3.6	. . 3 ⊢ (φ → (X +c 1c)FY)
73		df-br 4640	. . . 4 ⊢ ((X +c 1c)FY ↔ ⟨(X +c 1c), Y⟩ ∈ F)
74		snex 4111	. . . . 5 ⊢ {⟨0c, I⟩} ∈ V
75		csucex 6258	. . . . . 6 ⊢ (w ∈ V ↦ (w +c 1c)) ∈ V
76		pprodexg 5837	. . . . . 6 ⊢ (((w ∈ V ↦ (w +c 1c)) ∈ V ∧ G ∈ V) → PProd ((w ∈ V ↦ (w +c 1c)), G) ∈ V)
77	75, 49, 76	sylancr 644	. . . . 5 ⊢ (φ → PProd ((w ∈ V ↦ (w +c 1c)), G) ∈ V)
78		df-frec 6308	. . . . . . 7 ⊢ FRec (G, I) = Clos1 ({⟨0c, I⟩}, PProd ((w ∈ V ↦ (w +c 1c)), G))
79	48, 78	eqtri 2373	. . . . . 6 ⊢ F = Clos1 ({⟨0c, I⟩}, PProd ((w ∈ V ↦ (w +c 1c)), G))
80	79	clos1basesucg 5884	. . . . 5 ⊢ (({⟨0c, I⟩} ∈ V ∧ PProd ((w ∈ V ↦ (w +c 1c)), G) ∈ V) → (⟨(X +c 1c), Y⟩ ∈ F ↔ (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} ∨ ∃a ∈ F a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩)))
81	74, 77, 80	sylancr 644	. . . 4 ⊢ (φ → (⟨(X +c 1c), Y⟩ ∈ F ↔ (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} ∨ ∃a ∈ F a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩)))
82	73, 81	syl5bb 248	. . 3 ⊢ (φ → ((X +c 1c)FY ↔ (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} ∨ ∃a ∈ F a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩)))
83	72, 82	mpbid 201	. 2 ⊢ (φ → (⟨(X +c 1c), Y⟩ ∈ {⟨0c, I⟩} ∨ ∃a ∈ F a PProd ((w ∈ V ↦ (w +c 1c)), G)⟨(X +c 1c), Y⟩))
84	15, 71, 83	mpjaod 370	1 ⊢ (φ → ∃z(XFz ∧ zGY))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  {csn 3737  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +_c cplc 4375  ⟨cop 4561   class class class wbr 4639  dom cdm 4772  ran crn 4773   Fn wfn 4776   ‘cfv 4781   ↦ cmpt 5651   PProd cpprod 5737   Clos1 cclos1 5872   FRec cfrec 6307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-clos1 5873  df-frec 6308

This theorem is referenced by:  fnfrec  6318