Theorem fnlfvk3n0 36069

Description: The value of 𝐹_𝐿 when the value of 𝐾₃ is nonzero. Part of Proposition 15.14 of [TakeutiZaring] p. 158. (Contributed by BTernaryTau, 17-Oct-2026.)

Assertion

Ref	Expression
fnlfvk3n0	⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐹𝐿‘𝐴) = (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩))

Proof of Theorem fnlfvk3n0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fnl 36037	. . . 4 ⊢ 𝐹𝐿 = recs((𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩))))
2	1	tfr2 8444	. . 3 ⊢ (𝐴 ∈ On → (𝐹𝐿‘𝐴) = ((𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩)))‘(𝐹𝐿 ↾ 𝐴)))
3	2	adantr 486	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐹𝐿‘𝐴) = ((𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩)))‘(𝐹𝐿 ↾ 𝐴)))
4		eqid 2763	. . 3 ⊢ (𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩))) = (𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩)))
5		dmeq 5897	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹𝐿 ↾ 𝐴) → dom 𝑥 = dom (𝐹𝐿 ↾ 𝐴))
6	5	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom 𝑥 = dom (𝐹𝐿 ↾ 𝐴))
7		onss 7833	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8	1	tfr1 8443	. . . . . . . . . . . . . 14 ⊢ 𝐹𝐿 Fn On
9	8	fndmi 6673	. . . . . . . . . . . . 13 ⊢ dom 𝐹𝐿 = On
10	9	sseq2i 3942	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹𝐿 ↔ 𝐴 ⊆ On)
11		ssdmres 6021	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹𝐿 ↔ dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
12	10, 11	bitr3i 280	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On ↔ dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
13	7, 12	sylib 221	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
14	13	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
15	6, 14	eqtrd 2798	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom 𝑥 = 𝐴)
16	15	fveq2d 6920	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘dom 𝑥) = (𝐾3‘𝐴))
17		simplr 781	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘𝐴) ≠ ∅)
18	16, 17	eqnetrd 3026	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘dom 𝑥) ≠ ∅)
19	18	neneqd 2964	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → ¬ (𝐾3‘dom 𝑥) = ∅)
20	19	iffalsed 4496	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩)) = (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩))
21		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → 𝑥 = (𝐹𝐿 ↾ 𝐴))
22	15	fveq2d 6920	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾1‘dom 𝑥) = (𝐾1‘𝐴))
23	21, 22	fveq12d 6923	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾1‘dom 𝑥)) = ((𝐹𝐿 ↾ 𝐴)‘(𝐾1‘𝐴)))
24		k39o 36058	. . . . . . . . . . . 12 ⊢ (𝐾3‘𝐴) ∈ 9o
25		9on 35908	. . . . . . . . . . . . 13 ⊢ 9o ∈ On
26	25	oneli 6504	. . . . . . . . . . . 12 ⊢ ((𝐾3‘𝐴) ∈ 9o → (𝐾3‘𝐴) ∈ On)
27	24, 26	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐾3‘𝐴) ∈ On
28		on0eln0 6446	. . . . . . . . . . 11 ⊢ ((𝐾3‘𝐴) ∈ On → (∅ ∈ (𝐾3‘𝐴) ↔ (𝐾3‘𝐴) ≠ ∅))
29	27, 28	ax-mp 5	. . . . . . . . . 10 ⊢ (∅ ∈ (𝐾3‘𝐴) ↔ (𝐾3‘𝐴) ≠ ∅)
30		k3n0k1lt 36064	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐾1‘𝐴) ∈ 𝐴)
31	29, 30	sylan2br 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐾1‘𝐴) ∈ 𝐴)
32	31	fvresd 6936	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → ((𝐹𝐿 ↾ 𝐴)‘(𝐾1‘𝐴)) = (𝐹𝐿‘(𝐾1‘𝐴)))
33	32	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → ((𝐹𝐿 ↾ 𝐴)‘(𝐾1‘𝐴)) = (𝐹𝐿‘(𝐾1‘𝐴)))
34	23, 33	eqtrd 2798	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾1‘dom 𝑥)) = (𝐹𝐿‘(𝐾1‘𝐴)))
35	15	fveq2d 6920	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾2‘dom 𝑥) = (𝐾2‘𝐴))
36	21, 35	fveq12d 6923	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾2‘dom 𝑥)) = ((𝐹𝐿 ↾ 𝐴)‘(𝐾2‘𝐴)))
37		k3n0k2lt 36065	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐾2‘𝐴) ∈ 𝐴)
38	29, 37	sylan2br 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐾2‘𝐴) ∈ 𝐴)
39	38	fvresd 6936	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → ((𝐹𝐿 ↾ 𝐴)‘(𝐾2‘𝐴)) = (𝐹𝐿‘(𝐾2‘𝐴)))
40	39	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → ((𝐹𝐿 ↾ 𝐴)‘(𝐾2‘𝐴)) = (𝐹𝐿‘(𝐾2‘𝐴)))
41	36, 40	eqtrd 2798	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾2‘dom 𝑥)) = (𝐹𝐿‘(𝐾2‘𝐴)))
42	16, 34, 41	oteq123d 4871	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → ⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩ = ⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩)
43	42	fveq2d 6920	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩) = (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩))
44	20, 43	eqtrd 2798	. . 3 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩)) = (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩))
45		fnfun 6669	. . . . . 6 ⊢ (𝐹𝐿 Fn On → Fun 𝐹𝐿)
46	8, 45	ax-mp 5	. . . . 5 ⊢ Fun 𝐹𝐿
47		resfunexg 7252	. . . . 5 ⊢ ((Fun 𝐹𝐿 ∧ 𝐴 ∈ On) → (𝐹𝐿 ↾ 𝐴) ∈ V)
48	46, 47	mpan 703	. . . 4 ⊢ (𝐴 ∈ On → (𝐹𝐿 ↾ 𝐴) ∈ V)
49	48	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐹𝐿 ↾ 𝐴) ∈ V)
50		fvexd 6931	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩) ∈ V)
51	4, 44, 49, 50	fvmptd2 7033	. 2 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → ((𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩)))‘(𝐹𝐿 ↾ 𝐴)) = (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩))
52	3, 51	eqtrd 2798	1 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐹𝐿‘𝐴) = (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ifcif 4485  ⟨cotp 4863   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   ↾ cres 5660  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566  9_oc9o 35898  ℱ cgdlopc 35992  𝐾₁ck1 36025  𝐾₂ck2 36026  𝐾₃ck3 36027  𝐹_𝐿cfnl 36028

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-mpt3 7716  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-smo 8392  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-3o 8516  df-4o 8517  df-oadd 8518  df-omul 8519  df-oi 9541  df-5o 35899  df-6o 35900  df-7o 35901  df-8o 35902  df-9o 35903  df-lexo 36030  df-r0 36031  df-j0 36032  df-j 36033  df-k1 36034  df-k2 36035  df-k3 36036  df-fnl 36037

This theorem is used by: (None)