Theorem fnlfvk3n0 35979

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 35947	. . . 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 8414	. . 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 2760	. . 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 5886	. . . . . . . . . 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 7803	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8	1	tfr1 8413	. . . . . . . . . . . . . 14 ⊢ 𝐹𝐿 Fn On
9	8	fndmi 6648	. . . . . . . . . . . . 13 ⊢ dom 𝐹𝐿 = On
10	9	sseq2i 3938	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹𝐿 ↔ 𝐴 ⊆ On)
11		ssdmres 6006	. . . . . . . . . . . 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 2795	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom 𝑥 = 𝐴)
16	15	fveq2d 6894	. . . . . . 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 3022	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘dom 𝑥) ≠ ∅)
19	18	neneqd 2960	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → ¬ (𝐾3‘dom 𝑥) = ∅)
20	19	iffalsed 4492	. . . 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 6894	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾1‘dom 𝑥) = (𝐾1‘𝐴))
23	21, 22	fveq12d 6897	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾1‘dom 𝑥)) = ((𝐹𝐿 ↾ 𝐴)‘(𝐾1‘𝐴)))
24		k39o 35968	. . . . . . . . . . . 12 ⊢ (𝐾3‘𝐴) ∈ 9o
25		9on 35843	. . . . . . . . . . . . 13 ⊢ 9o ∈ On
26	25	oneli 6481	. . . . . . . . . . . 12 ⊢ ((𝐾3‘𝐴) ∈ 9o → (𝐾3‘𝐴) ∈ On)
27	24, 26	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐾3‘𝐴) ∈ On
28		on0eln0 6423	. . . . . . . . . . 11 ⊢ ((𝐾3‘𝐴) ∈ On → (∅ ∈ (𝐾3‘𝐴) ↔ (𝐾3‘𝐴) ≠ ∅))
29	27, 28	ax-mp 5	. . . . . . . . . 10 ⊢ (∅ ∈ (𝐾3‘𝐴) ↔ (𝐾3‘𝐴) ≠ ∅)
30		k3n0k1lt 35974	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐾1‘𝐴) ∈ 𝐴)
31	29, 30	sylan2br 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐾1‘𝐴) ∈ 𝐴)
32	31	fvresd 6910	. . . . . . . 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 2795	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾1‘dom 𝑥)) = (𝐹𝐿‘(𝐾1‘𝐴)))
35	15	fveq2d 6894	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾2‘dom 𝑥) = (𝐾2‘𝐴))
36	21, 35	fveq12d 6897	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾2‘dom 𝑥)) = ((𝐹𝐿 ↾ 𝐴)‘(𝐾2‘𝐴)))
37		k3n0k2lt 35975	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐾2‘𝐴) ∈ 𝐴)
38	29, 37	sylan2br 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐾2‘𝐴) ∈ 𝐴)
39	38	fvresd 6910	. . . . . . . 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 2795	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝑥‘(𝐾2‘dom 𝑥)) = (𝐹𝐿‘(𝐾2‘𝐴)))
42	16, 34, 41	oteq123d 4865	. . . . 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 6894	. . . 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 2795	. . 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 6644	. . . . . 6 ⊢ (𝐹𝐿 Fn On → Fun 𝐹𝐿)
46	8, 45	ax-mp 5	. . . . 5 ⊢ Fun 𝐹𝐿
47		resfunexg 7225	. . . . 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 6905	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩) ∈ V)
51	4, 44, 49, 50	fvmptd2 7007	. 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 2795	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 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ifcif 4481  ⟨cotp 4857   ↦ cmpt 5185  dom cdm 5651  ran crn 5652   ↾ cres 5653  Oncon0 6365  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  9_oc9o 35833  ℱ cgdlopc 35919  𝐾₁ck1 35935  𝐾₂ck2 35936  𝐾₃ck3 35937  𝐹_𝐿cfnl 35938

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mpt3 7688  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-smo 8362  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-3o 8486  df-4o 8487  df-oadd 8488  df-omul 8489  df-oi 9511  df-5o 35834  df-6o 35835  df-7o 35836  df-8o 35837  df-9o 35838  df-lexo 35940  df-r0 35941  df-j0 35942  df-j 35943  df-k1 35944  df-k2 35945  df-k3 35946  df-fnl 35947

This theorem is used by: (None)