Theorem onvf1odlem3 35543

Description: Lemma for onvf1od 35545. 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 8384	. 2 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)))
3	1	tfr1 8383	. . . . 5 ⊢ 𝐹 Fn On
4		fnfun 6635	. . . . 5 ⊢ (𝐹 Fn On → Fun 𝐹)
5	3, 4	ax-mp 5	. . . 4 ⊢ Fun 𝐹
6		resfunexg 7213	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ On) → (𝐹 ↾ 𝐴) ∈ V)
7	5, 6	mpan 702	. . 3 ⊢ (𝐴 ∈ On → (𝐹 ↾ 𝐴) ∈ V)
8		eleq1w 2844	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑣 → (𝑡 ∈ ran 𝑟 ↔ 𝑣 ∈ ran 𝑟))
9	8	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑣 → (¬ 𝑡 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ ran 𝑟))
10	9	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (¬ 𝑡 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ ran 𝑟))
11		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑢 → (𝑅1‘𝑠) = (𝑅1‘𝑢))
12	11	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑠 = 𝑢 ∧ 𝑡 = 𝑣) → (𝑅1‘𝑠) = (𝑅1‘𝑢))
13	10, 12	cbvrexdva2 3339	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑢 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟))
14	13	cbvrabv 3424	. . . . . . . . . . 11 ⊢ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟}
15		rneq 5926	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = ran (𝐹 ↾ 𝐴))
16		df-ima 5674	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
17	15, 16	eqtr4di 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ran 𝑟 = (𝐹 “ 𝐴))
18	17	eleq2d 2847	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑣 ∈ ran 𝑟 ↔ 𝑣 ∈ (𝐹 “ 𝐴)))
19	18	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (¬ 𝑣 ∈ ran 𝑟 ↔ ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
20	19	rexbidv 3187	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟 ↔ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)))
21	20	rabbidv 3421	. . . . . . . . . . 11 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
22	14, 21	eqtrid 2808	. . . . . . . . . 10 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)})
23	22	inteqd 4916	. . . . . . . . 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 2814	. . . . . . . 8 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟} = 𝐵)
26	25	fveq2d 6885	. . . . . . 7 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) = (𝑅1‘𝐵))
27	26, 17	difeq12d 4081	. . . . . 6 ⊢ (𝑟 = (𝐹 ↾ 𝐴) → ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟) = ((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))
28	27	fveq2d 6885	. . . . 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 2814	. . . 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 2844	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑡 → (𝑦 ∈ ran 𝑤 ↔ 𝑡 ∈ ran 𝑤))
34	33	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → (¬ 𝑦 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑤))
35	34	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (¬ 𝑦 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑤))
36		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑠 → (𝑅1‘𝑥) = (𝑅1‘𝑠))
37	36	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝑅1‘𝑥) = (𝑅1‘𝑠))
38	35, 37	cbvrexdva2 3339	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤))
39	38	cbvrabv 3424	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤}
40		rneq 5926	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑟 → ran 𝑤 = ran 𝑟)
41	40	eleq2d 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑟 → (𝑡 ∈ ran 𝑤 ↔ 𝑡 ∈ ran 𝑟))
42	41	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑟 → (¬ 𝑡 ∈ ran 𝑤 ↔ ¬ 𝑡 ∈ ran 𝑟))
43	42	rexbidv 3187	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑟 → (∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤 ↔ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟))
44	43	rabbidv 3421	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑟 → {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
45	39, 44	eqtrid 2808	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑟 → {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
46	45	inteqd 4916	. . . . . . . . . 10 ⊢ (𝑤 = 𝑟 → ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤} = ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
47	32, 46	eqtrid 2808	. . . . . . . . 9 ⊢ (𝑤 = 𝑟 → 𝑀 = ∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟})
48	47	fveq2d 6885	. . . . . . . 8 ⊢ (𝑤 = 𝑟 → (𝑅1‘𝑀) = (𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}))
49	48, 40	difeq12d 4081	. . . . . . 7 ⊢ (𝑤 = 𝑟 → ((𝑅1‘𝑀) ∖ ran 𝑤) = ((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟))
50	49	fveq2d 6885	. . . . . 6 ⊢ (𝑤 = 𝑟 → (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤)) = (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
51	31, 50	eqtrid 2808	. . . . 5 ⊢ (𝑤 = 𝑟 → 𝑁 = (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
52	51	cbvmptv 5214	. . . 4 ⊢ (𝑤 ∈ V ↦ 𝑁) = (𝑟 ∈ V ↦ (𝐺‘((𝑅1‘∩ {𝑠 ∈ On ∣ ∃𝑡 ∈ (𝑅1‘𝑠) ¬ 𝑡 ∈ ran 𝑟}) ∖ ran 𝑟)))
53	29	fvexi 6895	. . . 4 ⊢ 𝐶 ∈ V
54	30, 52, 53	fvmpt 6989	. . 3 ⊢ ((𝐹 ↾ 𝐴) ∈ V → ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)) = 𝐶)
55	7, 54	syl 18	. 2 ⊢ (𝐴 ∈ On → ((𝑤 ∈ V ↦ 𝑁)‘(𝐹 ↾ 𝐴)) = 𝐶)
56	2, 55	eqtrd 2796	1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  {crab 3414  Vcvv 3453   ∖ cdif 3901  ∩ cint 4911   ↦ cmpt 5191  ran crn 5662   ↾ cres 5663   “ cima 5664  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  recscrecs 8356  𝑅₁cr1 9733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357

This theorem is referenced by:  onvf1odlem4  35544  onvf1od  35545