Theorem onvf1odlem3 35403

Description: Lemma for onvf1od 35405. The value of 𝐹 at an ordinal 𝐴. (Contributed by BTernaryTau, 2-Dec-2025.)

Hypotheses

Ref	Expression
onvf1odlem3.1	⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}
onvf1odlem3.2	⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))
onvf1odlem3.3	⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))
onvf1odlem3.4	⊢ 𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)}
onvf1odlem3.5	⊢ 𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))

Assertion

Ref	Expression
onvf1odlem3	⊢ (𝐴 ∈ On → (𝐹‘𝐴) = 𝐶)

Distinct variable groups:   𝑤,𝐺   𝑢,𝐴,𝑣   𝑢,𝐹,𝑣   𝑥,𝑤,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥,𝑦,𝑤,𝑣,𝑢)   𝐶(𝑥,𝑦,𝑤,𝑣,𝑢)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑣,𝑢)   𝑀(𝑥,𝑦,𝑤,𝑣,𝑢)   𝑁(𝑥,𝑦,𝑤,𝑣,𝑢)

Proof of Theorem onvf1odlem3

Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onvf1odlem3.3	. . 3 ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))
2	1	tfr2 8357	. 2 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)))
3	1	tfr1 8356	. . . . 5 ⊢ 𝐹 Fn On
4		fnfun 6610	. . . . 5 ⊢ (𝐹 Fn On → Fun 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Fun 𝐹
6		resfunexg 7188	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ On) → (𝐹 ↾ 𝐴) ∈ V)
7	5, 6	mpan 698	. . 3 ⊢ (𝐴 ∈ On → (𝐹 ↾ 𝐴) ∈ V)
8		eleq1w 2839	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑣 → (𝑡 ∈ ran 𝑟 ↔ 𝑣 ∈ ran 𝑟))
9	8	notbid 320	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑣 → (¬ 𝑡 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ ran 𝑟))
10	9	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (¬ 𝑡 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ ran 𝑟))
11		fveq2 6856	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑢 → (𝑅1‘𝑠) = (𝑅1‘𝑢))
12	11	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (𝑅1‘𝑠) = (𝑅1‘𝑢))
13	10, 12	cbvrexdva2 3333	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑢 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟))
14	13	cbvrabv 3418	. . . . . . . . . . 11 ⊢ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟}
15		rneq 5905	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = ran (𝐹 ↾ 𝐴))
16		df-ima 5653	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
17	15, 16	eqtr4di 2809	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = (𝐹 “ 𝐴))
18	17	eleq2d 2842	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑣 ∈ ran 𝑟 ↔ 𝑣 ∈ (𝐹 “ 𝐴)))
19	18	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (¬ 𝑣 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
20	19	rexbidv 3180	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
21	20	rabbidv 3415	. . . . . . . . . . 11 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
22	14, 21	eqtrid 2803	. . . . . . . . . 10 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
23	22	inteqd 4904	. . . . . . . . 9 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
24		onvf1odlem3.4	. . . . . . . . 9 ⊢ 𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)}
25	23, 24	eqtr4di 2809	. . . . . . . 8 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = 𝐵)
26	25	fveq2d 6860	. . . . . . 7 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) = (𝑅1‘𝐵))
27	26, 17	difeq12d 4076	. . . . . 6 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟) = ((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))
28	27	fveq2d 6860	. . . . 5 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)) = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴))))
29		onvf1odlem3.5	. . . . 5 ⊢ 𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))
30	28, 29	eqtr4di 2809	. . . 4 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)) = 𝐶)
31		onvf1odlem3.2	. . . . . 6 ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))
32		onvf1odlem3.1	. . . . . . . . . 10 ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}
33		eleq1w 2839	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑡 → (𝑦 ∈ ran 𝑤 ↔ 𝑡 ∈ ran 𝑤))
34	33	notbid 320	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (¬ 𝑦 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑤))
35	34	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (¬ 𝑦 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑤))
36		fveq2 6856	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → (𝑅1‘𝑥) = (𝑅1‘𝑠))
37	36	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝑅1‘𝑥) = (𝑅1‘𝑠))
38	35, 37	cbvrexdva2 3333	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤))
39	38	cbvrabv 3418	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤}
40		rneq 5905	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑟 → ran 𝑤 = ran 𝑟)
41	40	eleq2d 2842	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑟 → (𝑡 ∈ ran 𝑤 ↔ 𝑡 ∈ ran 𝑟))
42	41	notbid 320	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑟 → (¬ 𝑡 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑟))
43	42	rexbidv 3180	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟))
44	43	rabbidv 3415	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
45	39, 44	eqtrid 2803	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑟 → {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
46	45	inteqd 4904	. . . . . . . . . 10 ⊢ (𝑤 = 𝑟 → ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
47	32, 46	eqtrid 2803	. . . . . . . . 9 ⊢ (𝑤 = 𝑟 → 𝑀 = ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
48	47	fveq2d 6860	. . . . . . . 8 ⊢ (𝑤 = 𝑟 → (𝑅1‘𝑀) = (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}))
49	48, 40	difeq12d 4076	. . . . . . 7 ⊢ (𝑤 = 𝑟 → ((𝑅1‘𝑀) ∖ ran 𝑤) = ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟))
50	49	fveq2d 6860	. . . . . 6 ⊢ (𝑤 = 𝑟 → (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤)) = (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
51	31, 50	eqtrid 2803	. . . . 5 ⊢ (𝑤 = 𝑟 → 𝑁 = (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
52	51	cbvmptv 5198	. . . 4 ⊢ (𝑤 ∈ V ↦ 𝑁) = (𝑟 ∈ V ↦ (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
53	29	fvexi 6870	. . . 4 ⊢ 𝐶 ∈ V
54	30, 52, 53	fvmpt 6964	. . 3 ⊢ ((𝐹 ↾ 𝐴) ∈ V → ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)) = 𝐶)
55	7, 54	syl 17	. 2 ⊢ (𝐴 ∈ On → ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)) = 𝐶)
56	2, 55	eqtrd 2791	1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1554   ∈ wcel 2136  ∃wrex 3080  {crab 3408  Vcvv 3448   ∖ cdif 3896  ∩ cint 4899   ↦ cmpt 5175  ran crn 5641   ↾ cres 5642   “ cima 5643  Oncon0 6335  Fun wfun 6504   Fn wfn 6505  ‘cfv 6510  recscrecs 8329  𝑅₁cr1 9710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330

This theorem is referenced by:  onvf1odlem4  35404  onvf1od  35405