Theorem depindlem2 16551

Description: Lemma for depind 16553. (Contributed by Matthew House, 14-Apr-2026.)

Hypotheses

Ref	Expression
depind.p	⊢ (𝜑 → 𝑃:ℕ0⟶V)
depind.0	⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
depind.h	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
depindlem1.4	⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))

Assertion

Ref	Expression
depindlem2	⊢ (𝜑 → 𝐹 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛))

Distinct variable groups:   ℎ,𝑛,𝑥   𝐴,𝑚,𝑛   𝑛,𝐹   𝑚,𝐻,𝑛   𝑃,𝑛

Allowed substitution hints:   𝜑(𝑥,ℎ,𝑚,𝑛)   𝐴(𝑥,ℎ)   𝑃(𝑥,ℎ,𝑚)   𝐹(𝑥,ℎ,𝑚)   𝐻(𝑥,ℎ)

Proof of Theorem depindlem2

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

Step	Hyp	Ref	Expression
1		depind.p	. . . . . 6 ⊢ (𝜑 → 𝑃:ℕ0⟶V)
2		depind.0	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
3		depind.h	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
4		depindlem1.4	. . . . . 6 ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))
5	1, 2, 3, 4	depindlem1 16550	. . . . 5 ⊢ (𝜑 → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))
6	5	simp1d 1036	. . . 4 ⊢ (𝜑 → 𝐹:ℕ0⟶V)
7		nn0ex 9507	. . . . 5 ⊢ ℕ0 ∈ V
8	7	a1i 9	. . . 4 ⊢ (𝜑 → ℕ0 ∈ V)
9	6, 8	fexd 5918	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
10	6	ffnd 5511	. . 3 ⊢ (𝜑 → 𝐹 Fn ℕ0)
11		fveq2 5672	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
12		fveq2 5672	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑃‘𝑦) = (𝑃‘0))
13	11, 12	eleq12d 2305	. . . . . . 7 ⊢ (𝑦 = 0 → ((𝐹‘𝑦) ∈ (𝑃‘𝑦) ↔ (𝐹‘0) ∈ (𝑃‘0)))
14	13	imbi2d 230	. . . . . 6 ⊢ (𝑦 = 0 → ((𝜑 → (𝐹‘𝑦) ∈ (𝑃‘𝑦)) ↔ (𝜑 → (𝐹‘0) ∈ (𝑃‘0))))
15		fveq2 5672	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
16		fveq2 5672	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (𝑃‘𝑦) = (𝑃‘𝑘))
17	15, 16	eleq12d 2305	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ((𝐹‘𝑦) ∈ (𝑃‘𝑦) ↔ (𝐹‘𝑘) ∈ (𝑃‘𝑘)))
18	17	imbi2d 230	. . . . . 6 ⊢ (𝑦 = 𝑘 → ((𝜑 → (𝐹‘𝑦) ∈ (𝑃‘𝑦)) ↔ (𝜑 → (𝐹‘𝑘) ∈ (𝑃‘𝑘))))
19		fveq2 5672	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (𝐹‘𝑦) = (𝐹‘(𝑘 + 1)))
20		fveq2 5672	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (𝑃‘𝑦) = (𝑃‘(𝑘 + 1)))
21	19, 20	eleq12d 2305	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ((𝐹‘𝑦) ∈ (𝑃‘𝑦) ↔ (𝐹‘(𝑘 + 1)) ∈ (𝑃‘(𝑘 + 1))))
22	21	imbi2d 230	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → ((𝜑 → (𝐹‘𝑦) ∈ (𝑃‘𝑦)) ↔ (𝜑 → (𝐹‘(𝑘 + 1)) ∈ (𝑃‘(𝑘 + 1)))))
23	5	simp2d 1037	. . . . . . 7 ⊢ (𝜑 → (𝐹‘0) = 𝐴)
24	23, 2	eqeltrd 2311	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) ∈ (𝑃‘0))
25	5	simp3d 1038	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)))
26		fvoveq1 6075	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
27		fveq2 5672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐻‘𝑛) = (𝐻‘𝑘))
28		fveq2 5672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
29	27, 28	fveq12d 5679	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛)‘(𝐹‘𝑛)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
30	26, 29	eqeq12d 2249	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ↔ (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘))))
31	30	rspccva 2922	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
32	25, 31	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
33	32	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝐹‘𝑘) ∈ (𝑃‘𝑘)) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
34		fveq2 5672	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
35		fvoveq1 6075	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
36	27, 34, 35	feq123d 5501	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)) ↔ (𝐻‘𝑘):(𝑃‘𝑘)⟶(𝑃‘(𝑘 + 1))))
37	36	rspccva 2922	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘):(𝑃‘𝑘)⟶(𝑃‘(𝑘 + 1)))
38	3, 37	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘):(𝑃‘𝑘)⟶(𝑃‘(𝑘 + 1)))
39	38	ffvelcdmda 5814	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝐹‘𝑘) ∈ (𝑃‘𝑘)) → ((𝐻‘𝑘)‘(𝐹‘𝑘)) ∈ (𝑃‘(𝑘 + 1)))
40	33, 39	eqeltrd 2311	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝐹‘𝑘) ∈ (𝑃‘𝑘)) → (𝐹‘(𝑘 + 1)) ∈ (𝑃‘(𝑘 + 1)))
41	40	exp31 364	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ ℕ0 → ((𝐹‘𝑘) ∈ (𝑃‘𝑘) → (𝐹‘(𝑘 + 1)) ∈ (𝑃‘(𝑘 + 1)))))
42	41	com12 30	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → ((𝐹‘𝑘) ∈ (𝑃‘𝑘) → (𝐹‘(𝑘 + 1)) ∈ (𝑃‘(𝑘 + 1)))))
43	42	a2d 26	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → ((𝜑 → (𝐹‘𝑘) ∈ (𝑃‘𝑘)) → (𝜑 → (𝐹‘(𝑘 + 1)) ∈ (𝑃‘(𝑘 + 1)))))
44	14, 18, 22, 18, 24, 43	nn0ind 9698	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (𝜑 → (𝐹‘𝑘) ∈ (𝑃‘𝑘)))
45	44	impcom 125	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ (𝑃‘𝑘))
46	45	ralrimiva 2617	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) ∈ (𝑃‘𝑘))
47		elixp2 6939	. . 3 ⊢ (𝐹 ∈ X𝑘 ∈ ℕ0 (𝑃‘𝑘) ↔ (𝐹 ∈ V ∧ 𝐹 Fn ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) ∈ (𝑃‘𝑘)))
48	9, 10, 46, 47	syl3anbrc 1208	. 2 ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ ℕ0 (𝑃‘𝑘))
49	34	cbvixpv 6953	. 2 ⊢ X𝑛 ∈ ℕ0 (𝑃‘𝑛) = X𝑘 ∈ ℕ0 (𝑃‘𝑘)
50	48, 49	eleqtrrdi 2328	1 ⊢ (𝜑 → 𝐹 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815  ifcif 3622   ↦ cmpt 4173   Fn wfn 5349  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052   ∈ cmpo 6054  Xcixp 6935  0cc0 8132  1c1 8133   + caddc 8135   − cmin 8449  ℕ₀cn0 9501  seqcseq 10816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-ixp 6936  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-inn 9243  df-n0 9502  df-z 9583  df-uz 9860  df-seqfrec 10817

This theorem is referenced by:  depind  16553