Theorem onvf1odlem3 35088

Description: Lemma for onvf1od 35090. 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 8320	. 2 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)))
3	1	tfr1 8319	. . . . 5 ⊢ 𝐹 Fn On
4		fnfun 6582	. . . . 5 ⊢ (𝐹 Fn On → Fun 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Fun 𝐹
6		resfunexg 7151	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ On) → (𝐹 ↾ 𝐴) ∈ V)
7	5, 6	mpan 690	. . 3 ⊢ (𝐴 ∈ On → (𝐹 ↾ 𝐴) ∈ V)
8		eleq1w 2811	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑣 → (𝑡 ∈ ran 𝑟 ↔ 𝑣 ∈ ran 𝑟))
9	8	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑣 → (¬ 𝑡 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ ran 𝑟))
10	9	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (¬ 𝑡 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ ran 𝑟))
11		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑢 → (𝑅1‘𝑠) = (𝑅1‘𝑢))
12	11	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (𝑅1‘𝑠) = (𝑅1‘𝑢))
13	10, 12	cbvrexdva2 3312	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑢 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟))
14	13	cbvrabv 3405	. . . . . . . . . . 11 ⊢ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟}
15		rneq 5878	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = ran (𝐹 ↾ 𝐴))
16		df-ima 5632	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
17	15, 16	eqtr4di 2782	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = (𝐹 “ 𝐴))
18	17	eleq2d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑣 ∈ ran 𝑟 ↔ 𝑣 ∈ (𝐹 “ 𝐴)))
19	18	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (¬ 𝑣 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
20	19	rexbidv 3153	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
21	20	rabbidv 3402	. . . . . . . . . . 11 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
22	14, 21	eqtrid 2776	. . . . . . . . . 10 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
23	22	inteqd 4901	. . . . . . . . 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 2782	. . . . . . . 8 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = 𝐵)
26	25	fveq2d 6826	. . . . . . 7 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) = (𝑅1‘𝐵))
27	26, 17	difeq12d 4078	. . . . . 6 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟) = ((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))
28	27	fveq2d 6826	. . . . 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 2782	. . . 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 2811	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑡 → (𝑦 ∈ ran 𝑤 ↔ 𝑡 ∈ ran 𝑤))
34	33	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (¬ 𝑦 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑤))
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (¬ 𝑦 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑤))
36		fveq2 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → (𝑅1‘𝑥) = (𝑅1‘𝑠))
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝑅1‘𝑥) = (𝑅1‘𝑠))
38	35, 37	cbvrexdva2 3312	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤))
39	38	cbvrabv 3405	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤}
40		rneq 5878	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑟 → ran 𝑤 = ran 𝑟)
41	40	eleq2d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑟 → (𝑡 ∈ ran 𝑤 ↔ 𝑡 ∈ ran 𝑟))
42	41	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑟 → (¬ 𝑡 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑟))
43	42	rexbidv 3153	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟))
44	43	rabbidv 3402	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
45	39, 44	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑟 → {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
46	45	inteqd 4901	. . . . . . . . . 10 ⊢ (𝑤 = 𝑟 → ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
47	32, 46	eqtrid 2776	. . . . . . . . 9 ⊢ (𝑤 = 𝑟 → 𝑀 = ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
48	47	fveq2d 6826	. . . . . . . 8 ⊢ (𝑤 = 𝑟 → (𝑅1‘𝑀) = (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}))
49	48, 40	difeq12d 4078	. . . . . . 7 ⊢ (𝑤 = 𝑟 → ((𝑅1‘𝑀) ∖ ran 𝑤) = ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟))
50	49	fveq2d 6826	. . . . . 6 ⊢ (𝑤 = 𝑟 → (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤)) = (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
51	31, 50	eqtrid 2776	. . . . 5 ⊢ (𝑤 = 𝑟 → 𝑁 = (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
52	51	cbvmptv 5196	. . . 4 ⊢ (𝑤 ∈ V ↦ 𝑁) = (𝑟 ∈ V ↦ (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
53	29	fvexi 6836	. . . 4 ⊢ 𝐶 ∈ V
54	30, 52, 53	fvmpt 6930	. . 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 2764	1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3394  Vcvv 3436   ∖ cdif 3900  ∩ cint 4896   ↦ cmpt 5173  ran crn 5620   ↾ cres 5621   “ cima 5622  Oncon0 6307  Fun wfun 6476   Fn wfn 6477  ‘cfv 6482  recscrecs 8293  𝑅₁cr1 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294

This theorem is referenced by:  onvf1odlem4  35089  onvf1od  35090