Theorem onvf1odlem4 35249

Description: Lemma for onvf1od 35250. If the range of 𝐹 does not exist, then it must equal the universe. (Contributed by BTernaryTau, 4-Dec-2025.)

Hypotheses

Ref	Expression
onvf1odlem4.1	⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
onvf1odlem4.2	⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}
onvf1odlem4.3	⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))
onvf1odlem4.4	⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))
onvf1odlem4.5	⊢ 𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}
onvf1odlem4.6	⊢ 𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡)))

Assertion

Ref	Expression
onvf1odlem4	⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))

Distinct variable groups:   𝑧,𝐵   𝑧,𝐺   𝑤,𝐺   𝑥,𝑤,𝑦   𝑡,𝐹,𝑧   𝜑,𝑡,𝑣   𝑢,𝐹,𝑣,𝑡

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

Proof of Theorem onvf1odlem4

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

Step	Hyp	Ref	Expression
1		eqv 3448	. . . 4 ⊢ (ran 𝐹 = V ↔ ∀𝑣 𝑣 ∈ ran 𝐹)
2		exnal 1828	. . . . 5 ⊢ (∃𝑣 ¬ 𝑣 ∈ ran 𝐹 ↔ ¬ ∀𝑣 𝑣 ∈ ran 𝐹)
3		onvf1odlem4.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))
4	3	tfr1 8326	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn On
5		fvelrnb 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn On → (𝑠 ∈ ran 𝐹 ↔ ∃𝑡 ∈ On (𝐹‘𝑡) = 𝑠))
6	4, 5	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ran 𝐹 ↔ ∃𝑡 ∈ On (𝐹‘𝑡) = 𝑠)
7		onvf1odlem4.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}
8		onvf1odlem4.3	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))
9		onvf1odlem4.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}
10		onvf1odlem4.6	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡)))
11	7, 8, 3, 9, 10	onvf1odlem3 35248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ On → (𝐹‘𝑡) = 𝐶)
12	11	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) = 𝐶)
13		fnfun 6590	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 Fn On → Fun 𝐹)
14	4, 13	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Fun 𝐹
15		vex 3442	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑡 ∈ V
16	15	funimaex 6578	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Fun 𝐹 → (𝐹 “ 𝑡) ∈ V)
17	14, 16	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 “ 𝑡) ∈ V
18		onvf1odlem4.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
19	18, 9, 10	onvf1odlem2 35247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐹 “ 𝑡) ∈ V → 𝐶 ∈ ((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡))))
20	17, 19	mpi 20	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐶 ∈ ((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡)))
21	20	eldifad 3911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐶 ∈ (𝑅1‘𝐵))
22	21	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → 𝐶 ∈ (𝑅1‘𝐵))
23	12, 22	eqeltrd 2834	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) ∈ (𝑅1‘𝐵))
24		rankr1ai 9708	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑡) ∈ (𝑅1‘𝐵) → (rank‘(𝐹‘𝑡)) ∈ 𝐵)
25		onvf1odlem1 35246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 “ 𝑡) ∈ V → ∃𝑢 ∈ On ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡))
26	17, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∃𝑢 ∈ On ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)
27		onintrab2 7740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑢 ∈ On ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡) ↔ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} ∈ On)
28	9	eleq1i 2825	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐵 ∈ On ↔ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} ∈ On)
29	27, 28	bitr4i 278	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑢 ∈ On ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡) ↔ 𝐵 ∈ On)
30	26, 29	mpbi 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ On
31	30	oneli 6430	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → (rank‘(𝐹‘𝑡)) ∈ On)
32		fveq2 6832	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (rank‘(𝐹‘𝑡)) → (𝑅1‘𝑢) = (𝑅1‘(rank‘(𝐹‘𝑡))))
33	32	rexeqdv 3295	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (rank‘(𝐹‘𝑡)) → (∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡) ↔ ∃𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡))) ¬ 𝑣 ∈ (𝐹 “ 𝑡)))
34	33	onnminsb 7742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((rank‘(𝐹‘𝑡)) ∈ On → ((rank‘(𝐹‘𝑡)) ∈ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} → ¬ ∃𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡))) ¬ 𝑣 ∈ (𝐹 “ 𝑡)))
35	9	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 ↔ (rank‘(𝐹‘𝑡)) ∈ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)})
36		dfral2 3085	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡) ↔ ¬ ∃𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡))) ¬ 𝑣 ∈ (𝐹 “ 𝑡))
37	34, 35, 36	3imtr4g 296	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rank‘(𝐹‘𝑡)) ∈ On → ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡)))
38	31, 37	mpcom 38	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡))
39		imassrn 6028	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 “ 𝑡) ⊆ ran 𝐹
40	39	sseli 3927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ (𝐹 “ 𝑡) → 𝑣 ∈ ran 𝐹)
41	40	ralimi 3071	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡) → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹)
42	38, 41	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹)
43	23, 24, 42	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹)
44		2fveq3 6837	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑡) = 𝑠 → (𝑅1‘(rank‘(𝐹‘𝑡))) = (𝑅1‘(rank‘𝑠)))
45	44	raleqdv 3294	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑡) = 𝑠 → (∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹 ↔ ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
46	43, 45	syl5ibcom 245	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ((𝐹‘𝑡) = 𝑠 → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
47	46	rexlimdva 3135	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑡 ∈ On (𝐹‘𝑡) = 𝑠 → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
48	6, 47	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑠 ∈ ran 𝐹 → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
49	48	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹)
50		df-ral 3050	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹 ↔ ∀𝑣(𝑣 ∈ (𝑅1‘(rank‘𝑠)) → 𝑣 ∈ ran 𝐹))
51	49, 50	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → ∀𝑣(𝑣 ∈ (𝑅1‘(rank‘𝑠)) → 𝑣 ∈ ran 𝐹))
52	51	19.21bi 2194	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → (𝑣 ∈ (𝑅1‘(rank‘𝑠)) → 𝑣 ∈ ran 𝐹))
53	52	con3d 152	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → (¬ 𝑣 ∈ ran 𝐹 → ¬ 𝑣 ∈ (𝑅1‘(rank‘𝑠))))
54		rankon 9705	. . . . . . . . . . . 12 ⊢ (rank‘𝑠) ∈ On
55		vex 3442	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
56	55	ssrankr1 9745	. . . . . . . . . . . 12 ⊢ ((rank‘𝑠) ∈ On → ((rank‘𝑠) ⊆ (rank‘𝑣) ↔ ¬ 𝑣 ∈ (𝑅1‘(rank‘𝑠))))
57	54, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ ((rank‘𝑠) ⊆ (rank‘𝑣) ↔ ¬ 𝑣 ∈ (𝑅1‘(rank‘𝑠)))
58	53, 57	imbitrrdi 252	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → (¬ 𝑣 ∈ ran 𝐹 → (rank‘𝑠) ⊆ (rank‘𝑣)))
59	58	impancom 451	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑣 ∈ ran 𝐹) → (𝑠 ∈ ran 𝐹 → (rank‘𝑠) ⊆ (rank‘𝑣)))
60	59	ralrimiv 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑣 ∈ ran 𝐹) → ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣))
61		rankon 9705	. . . . . . . . 9 ⊢ (rank‘𝑣) ∈ On
62		sseq2 3958	. . . . . . . . . . 11 ⊢ (𝑟 = (rank‘𝑣) → ((rank‘𝑠) ⊆ 𝑟 ↔ (rank‘𝑠) ⊆ (rank‘𝑣)))
63	62	ralbidv 3157	. . . . . . . . . 10 ⊢ (𝑟 = (rank‘𝑣) → (∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟 ↔ ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣)))
64	63	rspcev 3574	. . . . . . . . 9 ⊢ (((rank‘𝑣) ∈ On ∧ ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣)) → ∃𝑟 ∈ On ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟)
65	61, 64	mpan 690	. . . . . . . 8 ⊢ (∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣) → ∃𝑟 ∈ On ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟)
66		bndrank 9751	. . . . . . . 8 ⊢ (∃𝑟 ∈ On ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟 → ran 𝐹 ∈ V)
67	60, 65, 66	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑣 ∈ ran 𝐹) → ran 𝐹 ∈ V)
68	67	expcom 413	. . . . . 6 ⊢ (¬ 𝑣 ∈ ran 𝐹 → (𝜑 → ran 𝐹 ∈ V))
69	68	exlimiv 1931	. . . . 5 ⊢ (∃𝑣 ¬ 𝑣 ∈ ran 𝐹 → (𝜑 → ran 𝐹 ∈ V))
70	2, 69	sylbir 235	. . . 4 ⊢ (¬ ∀𝑣 𝑣 ∈ ran 𝐹 → (𝜑 → ran 𝐹 ∈ V))
71	1, 70	sylnbi 330	. . 3 ⊢ (¬ ran 𝐹 = V → (𝜑 → ran 𝐹 ∈ V))
72	71	com12 32	. 2 ⊢ (𝜑 → (¬ ran 𝐹 = V → ran 𝐹 ∈ V))
73	72	con1d 145	1 ⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  {crab 3397  Vcvv 3438   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4283  ∩ cint 4900   ↦ cmpt 5177  ran crn 5623   “ cima 5625  Oncon0 6315  Fun wfun 6484   Fn wfn 6485  ‘cfv 6490  recscrecs 8300  𝑅₁cr1 9672  rankcrnk 9673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9674  df-rank 9675

This theorem is referenced by:  onvf1od  35250