Theorem onvf1odlem4 35598

Description: Lemma for onvf1od 35599. 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 3465	. . . 4 ⊢ (ran 𝐹 = V ↔ ∀𝑣 𝑣 ∈ ran 𝐹)
2		exnal 1857	. . . . 5 ⊢ (∃𝑣 ¬ 𝑣 ∈ ran 𝐹 ↔ ¬ ∀𝑣 𝑣 ∈ ran 𝐹)
3		onvf1odlem4.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))
4	3	tfr1 8380	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 Fn On
5		fvelrnb 6941	. . . . . . . . . . . . . . . . 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 35597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ On → (𝐹‘𝑡) = 𝐶)
12	11	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) = 𝐶)
13		fnfun 6635	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 Fn On → Fun 𝐹)
14	4, 13	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Fun 𝐹
15		vex 3459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑡 ∈ V
16	15	funimaex 6623	. . . . . . . . . . . . . . . . . . . . . . . 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 35596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐹 “ 𝑡) ∈ V → 𝐶 ∈ ((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡))))
20	17, 19	mpi 21	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐶 ∈ ((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡)))
21	20	eldifad 3917	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐶 ∈ (𝑅1‘𝐵))
22	21	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → 𝐶 ∈ (𝑅1‘𝐵))
23	12, 22	eqeltrd 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) ∈ (𝑅1‘𝐵))
24		rankr1ai 9766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑡) ∈ (𝑅1‘𝐵) → (rank‘(𝐹‘𝑡)) ∈ 𝐵)
25		onvf1odlem1 35595	. . . . . . . . . . . . . . . . . . . . . . . 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 7792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃𝑢 ∈ On ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡) ↔ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} ∈ On)
28	9	eleq1i 2854	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐵 ∈ On ↔ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} ∈ On)
29	27, 28	bitr4i 281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑢 ∈ On ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡) ↔ 𝐵 ∈ On)
30	26, 29	mpbi 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ On
31	30	oneli 6476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → (rank‘(𝐹‘𝑡)) ∈ On)
32		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = (rank‘(𝐹‘𝑡)) → (𝑅1‘𝑢) = (𝑅1‘(rank‘(𝐹‘𝑡))))
33	32	rexeqdv 3324	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = (rank‘(𝐹‘𝑡)) → (∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡) ↔ ∃𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡))) ¬ 𝑣 ∈ (𝐹 “ 𝑡)))
34	33	onnminsb 7794	. . . . . . . . . . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 ↔ (rank‘(𝐹‘𝑡)) ∈ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)})
36		dfral2 3116	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡) ↔ ¬ ∃𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡))) ¬ 𝑣 ∈ (𝐹 “ 𝑡))
37	34, 35, 36	3imtr4g 299	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((rank‘(𝐹‘𝑡)) ∈ On → ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡)))
38	31, 37	mpcom 39	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡))
39		imassrn 6073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 “ 𝑡) ⊆ ran 𝐹
40	39	sseli 3933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ (𝐹 “ 𝑡) → 𝑣 ∈ ran 𝐹)
41	40	ralimi 3102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ (𝐹 “ 𝑡) → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹)
42	38, 41	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘(𝐹‘𝑡)) ∈ 𝐵 → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹)
43	23, 24, 42	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹)
44		2fveq3 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑡) = 𝑠 → (𝑅1‘(rank‘(𝐹‘𝑡))) = (𝑅1‘(rank‘𝑠)))
45	44	raleqdv 3323	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑡) = 𝑠 → (∀𝑣 ∈ (𝑅1‘(rank‘(𝐹‘𝑡)))𝑣 ∈ ran 𝐹 ↔ ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
46	43, 45	syl5ibcom 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ((𝐹‘𝑡) = 𝑠 → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
47	46	rexlimdva 3166	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑡 ∈ On (𝐹‘𝑡) = 𝑠 → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
48	6, 47	biimtrid 245	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑠 ∈ ran 𝐹 → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹))
49	48	imp 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → ∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹)
50		df-ral 3080	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ (𝑅1‘(rank‘𝑠))𝑣 ∈ ran 𝐹 ↔ ∀𝑣(𝑣 ∈ (𝑅1‘(rank‘𝑠)) → 𝑣 ∈ ran 𝐹))
51	49, 50	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → ∀𝑣(𝑣 ∈ (𝑅1‘(rank‘𝑠)) → 𝑣 ∈ ran 𝐹))
52	51	19.21bi 2225	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → (𝑣 ∈ (𝑅1‘(rank‘𝑠)) → 𝑣 ∈ ran 𝐹))
53	52	con3d 153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → (¬ 𝑣 ∈ ran 𝐹 → ¬ 𝑣 ∈ (𝑅1‘(rank‘𝑠))))
54		rankon 9763	. . . . . . . . . . . 12 ⊢ (rank‘𝑠) ∈ On
55		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
56	55	ssrankr1 9803	. . . . . . . . . . . 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 255	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐹) → (¬ 𝑣 ∈ ran 𝐹 → (rank‘𝑠) ⊆ (rank‘𝑣)))
59	58	impancom 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑣 ∈ ran 𝐹) → (𝑠 ∈ ran 𝐹 → (rank‘𝑠) ⊆ (rank‘𝑣)))
60	59	ralrimiv 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑣 ∈ ran 𝐹) → ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣))
61		rankon 9763	. . . . . . . . 9 ⊢ (rank‘𝑣) ∈ On
62		sseq2 3963	. . . . . . . . . . 11 ⊢ (𝑟 = (rank‘𝑣) → ((rank‘𝑠) ⊆ 𝑟 ↔ (rank‘𝑠) ⊆ (rank‘𝑣)))
63	62	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑟 = (rank‘𝑣) → (∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟 ↔ ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣)))
64	63	rspcev 3581	. . . . . . . . 9 ⊢ (((rank‘𝑣) ∈ On ∧ ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣)) → ∃𝑟 ∈ On ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟)
65	61, 64	mpan 702	. . . . . . . 8 ⊢ (∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ (rank‘𝑣) → ∃𝑟 ∈ On ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟)
66		bndrank 9809	. . . . . . . 8 ⊢ (∃𝑟 ∈ On ∀𝑠 ∈ ran 𝐹(rank‘𝑠) ⊆ 𝑟 → ran 𝐹 ∈ V)
67	60, 65, 66	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑣 ∈ ran 𝐹) → ran 𝐹 ∈ V)
68	67	expcom 418	. . . . . 6 ⊢ (¬ 𝑣 ∈ ran 𝐹 → (𝜑 → ran 𝐹 ∈ V))
69	68	exlimiv 1960	. . . . 5 ⊢ (∃𝑣 ¬ 𝑣 ∈ ran 𝐹 → (𝜑 → ran 𝐹 ∈ V))
70	2, 69	sylbir 238	. . . 4 ⊢ (¬ ∀𝑣 𝑣 ∈ ran 𝐹 → (𝜑 → ran 𝐹 ∈ V))
71	1, 70	sylnbi 333	. . 3 ⊢ (¬ ran 𝐹 = V → (𝜑 → ran 𝐹 ∈ V))
72	71	com12 33	. 2 ⊢ (𝜑 → (¬ ran 𝐹 = V → ran 𝐹 ∈ V))
73	72	con1d 146	1 ⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  ∩ cint 4912   ↦ cmpt 5192  ran crn 5662   “ cima 5664  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  recscrecs 8353  𝑅₁cr1 9730  rankcrnk 9731

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  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-lim 6365  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-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-r1 9732  df-rank 9733

This theorem is used by:  onvf1od  35599