Theorem fnlfvk30 35978

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

Assertion

Ref	Expression
fnlfvk30	⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) → (𝐹𝐿‘𝐴) = (𝐹𝐿 “ 𝐴))

Proof of Theorem fnlfvk30

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	. . . . . . . . 9 ⊢ (𝑥 = (𝐹𝐿 ↾ 𝐴) → dom 𝑥 = dom (𝐹𝐿 ↾ 𝐴))
6	5	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom 𝑥 = dom (𝐹𝐿 ↾ 𝐴))
7		onss 7803	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8	1	tfr1 8413	. . . . . . . . . . . . 13 ⊢ 𝐹𝐿 Fn On
9	8	fndmi 6648	. . . . . . . . . . . 12 ⊢ dom 𝐹𝐿 = On
10	9	sseq2i 3938	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ dom 𝐹𝐿 ↔ 𝐴 ⊆ On)
11		ssdmres 6006	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ dom 𝐹𝐿 ↔ dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
12	10, 11	bitr3i 280	. . . . . . . . . 10 ⊢ (𝐴 ⊆ On ↔ dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
13	7, 12	sylib 221	. . . . . . . . 9 ⊢ (𝐴 ∈ On → dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
14	13	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom (𝐹𝐿 ↾ 𝐴) = 𝐴)
15	6, 14	eqtrd 2795	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom 𝑥 = 𝐴)
16	15	fveq2d 6894	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘dom 𝑥) = (𝐾3‘𝐴))
17		simplr 781	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘𝐴) = ∅)
18	16, 17	eqtrd 2795	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘dom 𝑥) = ∅)
19	18	iftrued 4489	. . . 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))⟩)) = ran 𝑥)
20		rneq 5919	. . . . . 6 ⊢ (𝑥 = (𝐹𝐿 ↾ 𝐴) → ran 𝑥 = ran (𝐹𝐿 ↾ 𝐴))
21		df-ima 5664	. . . . . 6 ⊢ (𝐹𝐿 “ 𝐴) = ran (𝐹𝐿 ↾ 𝐴)
22	20, 21	eqtr4di 2813	. . . . 5 ⊢ (𝑥 = (𝐹𝐿 ↾ 𝐴) → ran 𝑥 = (𝐹𝐿 “ 𝐴))
23	22	adantl 487	. . . 4 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → ran 𝑥 = (𝐹𝐿 “ 𝐴))
24	19, 23	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))⟩)) = (𝐹𝐿 “ 𝐴))
25		fnfun 6644	. . . . . 6 ⊢ (𝐹𝐿 Fn On → Fun 𝐹𝐿)
26	8, 25	ax-mp 5	. . . . 5 ⊢ Fun 𝐹𝐿
27		resfunexg 7225	. . . . 5 ⊢ ((Fun 𝐹𝐿 ∧ 𝐴 ∈ On) → (𝐹𝐿 ↾ 𝐴) ∈ V)
28	26, 27	mpan 703	. . . 4 ⊢ (𝐴 ∈ On → (𝐹𝐿 ↾ 𝐴) ∈ V)
29	28	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) → (𝐹𝐿 ↾ 𝐴) ∈ V)
30		funimaexg 6631	. . . . 5 ⊢ ((Fun 𝐹𝐿 ∧ 𝐴 ∈ On) → (𝐹𝐿 “ 𝐴) ∈ V)
31	26, 30	mpan 703	. . . 4 ⊢ (𝐴 ∈ On → (𝐹𝐿 “ 𝐴) ∈ V)
32	31	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) → (𝐹𝐿 “ 𝐴) ∈ V)
33	4, 24, 29, 32	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)) = (𝐹𝐿 “ 𝐴))
34	3, 33	eqtrd 2795	1 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) → (𝐹𝐿‘𝐴) = (𝐹𝐿 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ifcif 4481  ⟨cotp 4857   ↦ cmpt 5185  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Oncon0 6365  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  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-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-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-uni 4867  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-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-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-ov 7427  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-fnl 35947

This theorem is used by: (None)