Theorem fnlfvk30 36068

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 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	. . . . . . . . 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 7833	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8	1	tfr1 8443	. . . . . . . . . . . . 13 ⊢ 𝐹𝐿 Fn On
9	8	fndmi 6673	. . . . . . . . . . . 12 ⊢ dom 𝐹𝐿 = On
10	9	sseq2i 3942	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ dom 𝐹𝐿 ↔ 𝐴 ⊆ On)
11		ssdmres 6021	. . . . . . . . . . 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 2798	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → dom 𝑥 = 𝐴)
16	15	fveq2d 6920	. . . . . 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 2798	. . . . 5 ⊢ (((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) ∧ 𝑥 = (𝐹𝐿 ↾ 𝐴)) → (𝐾3‘dom 𝑥) = ∅)
19	18	iftrued 4493	. . . 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 5930	. . . . . 6 ⊢ (𝑥 = (𝐹𝐿 ↾ 𝐴) → ran 𝑥 = ran (𝐹𝐿 ↾ 𝐴))
21		df-ima 5671	. . . . . 6 ⊢ (𝐹𝐿 “ 𝐴) = ran (𝐹𝐿 ↾ 𝐴)
22	20, 21	eqtr4di 2816	. . . . 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 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))⟩)) = (𝐹𝐿 “ 𝐴))
25		fnfun 6669	. . . . . 6 ⊢ (𝐹𝐿 Fn On → Fun 𝐹𝐿)
26	8, 25	ax-mp 5	. . . . 5 ⊢ Fun 𝐹𝐿
27		resfunexg 7252	. . . . 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 6656	. . . . 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 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)) = (𝐹𝐿 “ 𝐴))
34	3, 33	eqtrd 2798	1 ⊢ ((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) → (𝐹𝐿‘𝐴) = (𝐹𝐿 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ifcif 4485  ⟨cotp 4863   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566  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-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-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-uni 4873  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-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-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-ov 7454  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-fnl 36037

This theorem is used by: (None)