Theorem onvf1odlem3 35085

Description: Lemma for onvf1od 35087. 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 8343	. 2 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)))
3	1	tfr1 8342	. . . . 5 ⊢ 𝐹 Fn On
4		fnfun 6600	. . . . 5 ⊢ (𝐹 Fn On → Fun 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Fun 𝐹
6		resfunexg 7171	. . . 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 6840	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑢 → (𝑅1‘𝑠) = (𝑅1‘𝑢))
12	11	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (𝑅1‘𝑠) = (𝑅1‘𝑢))
13	10, 12	cbvrexdva2 3319	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑢 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟))
14	13	cbvrabv 3413	. . . . . . . . . . 11 ⊢ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟}
15		rneq 5889	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = ran (𝐹 ↾ 𝐴))
16		df-ima 5644	. . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
21	20	rabbidv 3410	. . . . . . . . . . 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 4911	. . . . . . . . 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 6844	. . . . . . 7 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) = (𝑅1‘𝐵))
27	26, 17	difeq12d 4086	. . . . . 6 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟) = ((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))
28	27	fveq2d 6844	. . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → (𝑅1‘𝑥) = (𝑅1‘𝑠))
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝑅1‘𝑥) = (𝑅1‘𝑠))
38	35, 37	cbvrexdva2 3319	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤))
39	38	cbvrabv 3413	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤}
40		rneq 5889	. . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟))
44	43	rabbidv 3410	. . . . . . . . . . . 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 4911	. . . . . . . . . 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 6844	. . . . . . . 8 ⊢ (𝑤 = 𝑟 → (𝑅1‘𝑀) = (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}))
49	48, 40	difeq12d 4086	. . . . . . 7 ⊢ (𝑤 = 𝑟 → ((𝑅1‘𝑀) ∖ ran 𝑤) = ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟))
50	49	fveq2d 6844	. . . . . 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 5206	. . . 4 ⊢ (𝑤 ∈ V ↦ 𝑁) = (𝑟 ∈ V ↦ (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
53	29	fvexi 6854	. . . 4 ⊢ 𝐶 ∈ V
54	30, 52, 53	fvmpt 6950	. . 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 3402  Vcvv 3444   ∖ cdif 3908  ∩ cint 4906   ↦ cmpt 5183  ran crn 5632   ↾ cres 5633   “ cima 5634  Oncon0 6320  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  recscrecs 8316  𝑅₁cr1 9691

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 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317

This theorem is referenced by:  onvf1odlem4  35086  onvf1od  35087