Theorem onvfowev 35624

Description: If 𝐹 maps the ordinals onto the universe, then 𝑅 well-orders the universe. This is the ZFC version of (8 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋(𝑋 ≠ ∅ → ∃𝐹𝐹:On–onto→𝑋), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 12-Jun-2026.)

Hypotheses

Ref	Expression
onvfowev.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐻‘𝑥) ∈ (𝐻‘𝑦)}
onvfowev.2	⊢ 𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧}))

Assertion

Ref	Expression
onvfowev	⊢ (𝐹:On–onto→V → 𝑅 We V)

Distinct variable groups:   𝑥,𝐻,𝑦   𝑧,𝐹

Allowed substitution hints:   𝑅(𝑥, 𝑦, 𝑧)   𝐹(𝑥, 𝑦)   𝐻(𝑧)

Proof of Theorem onvfowev

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvimass 6089	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
2		fofn 6801	. . . . . . . 8 ⊢ (𝐹:On–onto→V → 𝐹 Fn On)
3	2	fndmd 6647	. . . . . . 7 ⊢ (𝐹:On–onto→V → dom 𝐹 = On)
4	1, 3	sseqtrid 3982	. . . . . 6 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑧}) ⊆ On)
5		vex 3462	. . . . . . . 8 ⊢ 𝑧 ∈ V
6		forn 6802	. . . . . . . 8 ⊢ (𝐹:On–onto→V → ran 𝐹 = V)
7	5, 6	eleqtrrid 2873	. . . . . . 7 ⊢ (𝐹:On–onto→V → 𝑧 ∈ ran 𝐹)
8		inisegn0 6105	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑧}) ≠ ∅)
9	7, 8	sylib 221	. . . . . 6 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑧}) ≠ ∅)
10		oninton 7803	. . . . . 6 ⊢ (((◡𝐹 “ {𝑧}) ⊆ On ∧ (◡𝐹 “ {𝑧}) ≠ ∅) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
11	4, 9, 10	syl2anc 596	. . . . 5 ⊢ (𝐹:On–onto→V → ∩ (◡𝐹 “ {𝑧}) ∈ On)
12	11	adantr 486	. . . 4 ⊢ ((𝐹:On–onto→V ∧ 𝑧 ∈ V) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
13		onvfowev.2	. . . 4 ⊢ 𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧}))
14	12, 13	fmptd 7116	. . 3 ⊢ (𝐹:On–onto→V → 𝐻:V⟶On)
15		fofun 6800	. . . . . 6 ⊢ (𝐹:On–onto→V → Fun 𝐹)
16		fvexd 6903	. . . . . . . . 9 ⊢ ((𝐹:On–onto→V ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → (𝐻‘𝑣) ∈ V)
17		vex 3462	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐹:On–onto→V → 𝑤 ∈ V)
19	11	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑤) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
20		sneq 4604	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
21	20	imaeq2d 6067	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (◡𝐹 “ {𝑧}) = (◡𝐹 “ {𝑤}))
22	21	inteqd 4922	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑤}))
23	22	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑤) → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑤}))
24	18, 19, 23	fvmptdv2 7015	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → (𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧})) → (𝐻‘𝑤) = ∩ (◡𝐹 “ {𝑤})))
25	13, 24	mpi 21	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → (𝐻‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
26		cnvimass 6089	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
27	26, 3	sseqtrid 3982	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑤}) ⊆ On)
28	17, 6	eleqtrrid 2873	. . . . . . . . . . . . . . 15 ⊢ (𝐹:On–onto→V → 𝑤 ∈ ran 𝐹)
29		inisegn0 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
30	28, 29	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑤}) ≠ ∅)
31		onint 7798	. . . . . . . . . . . . . 14 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
32	27, 30, 31	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
33	25, 32	eqeltrd 2866	. . . . . . . . . . . 12 ⊢ (𝐹:On–onto→V → (𝐻‘𝑤) ∈ (◡𝐹 “ {𝑤}))
34		eleq1 2854	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑣) = (𝐻‘𝑤) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤}) ↔ (𝐻‘𝑤) ∈ (◡𝐹 “ {𝑤})))
35	33, 34	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
36		vex 3462	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → 𝑣 ∈ V)
38	11	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑣) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
39		sneq 4604	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → {𝑧} = {𝑣})
40	39	imaeq2d 6067	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → (◡𝐹 “ {𝑧}) = (◡𝐹 “ {𝑣}))
41	40	inteqd 4922	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑣 → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑣}))
42	41	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑣) → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑣}))
43	37, 38, 42	fvmptdv2 7015	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → (𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧})) → (𝐻‘𝑣) = ∩ (◡𝐹 “ {𝑣})))
44	13, 43	mpi 21	. . . . . . . . . . . 12 ⊢ (𝐹:On–onto→V → (𝐻‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
45		cnvimass 6089	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
46	45, 3	sseqtrid 3982	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑣}) ⊆ On)
47	36, 6	eleqtrrid 2873	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → 𝑣 ∈ ran 𝐹)
48		inisegn0 6105	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
49	47, 48	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑣}) ≠ ∅)
50		onint 7798	. . . . . . . . . . . . 13 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
51	46, 49, 50	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝐹:On–onto→V → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
52	44, 51	eqeltrd 2866	. . . . . . . . . . 11 ⊢ (𝐹:On–onto→V → (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}))
53	35, 52	jctild 535	. . . . . . . . . 10 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤}))))
54	53	imp 412	. . . . . . . . 9 ⊢ ((𝐹:On–onto→V ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
55		eleq1 2854	. . . . . . . . . 10 ⊢ (𝑢 = (𝐻‘𝑣) → (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣})))
56		eleq1 2854	. . . . . . . . . 10 ⊢ (𝑢 = (𝐻‘𝑣) → (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
57	55, 56	anbi12d 644	. . . . . . . . 9 ⊢ (𝑢 = (𝐻‘𝑣) → ((𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤})) ↔ ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤}))))
58	16, 54, 57	spcedv 3560	. . . . . . . 8 ⊢ ((𝐹:On–onto→V ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → ∃𝑢(𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤})))
59	58	ex 418	. . . . . . 7 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → ∃𝑢(𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤}))))
60		elinisegg 6100	. . . . . . . . . 10 ⊢ ((𝑣 ∈ V ∧ 𝑢 ∈ V) → (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ 𝑢𝐹𝑣))
61	60	el2v 3465	. . . . . . . . 9 ⊢ (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ 𝑢𝐹𝑣)
62		elinisegg 6100	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑢 ∈ V) → (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ 𝑢𝐹𝑤))
63	62	el2v 3465	. . . . . . . . 9 ⊢ (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ 𝑢𝐹𝑤)
64	61, 63	anbi12i 640	. . . . . . . 8 ⊢ ((𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤})) ↔ (𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤))
65	64	exbii 1881	. . . . . . 7 ⊢ (∃𝑢(𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤})) ↔ ∃𝑢(𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤))
66	59, 65	imbitrdi 254	. . . . . 6 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → ∃𝑢(𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤)))
67		funeu 6568	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣) → ∃!𝑣 𝑢𝐹𝑣)
68	67	3adant3 1150	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → ∃!𝑣 𝑢𝐹𝑣)
69		3simpc 1168	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → (𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤))
70		breq2 5118	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝑢𝐹𝑣 ↔ 𝑢𝐹𝑤))
71	70	eu4 2646	. . . . . . . . . . 11 ⊢ (∃!𝑣 𝑢𝐹𝑣 ↔ (∃𝑣 𝑢𝐹𝑣 ∧ ∀𝑣∀𝑤((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤)))
72	71	simprbi 503	. . . . . . . . . 10 ⊢ (∃!𝑣 𝑢𝐹𝑣 → ∀𝑣∀𝑤((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
73	72	19.21bbi 2229	. . . . . . . . 9 ⊢ (∃!𝑣 𝑢𝐹𝑣 → ((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
74	68, 69, 73	sylc 66	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤)
75	74	3expib 1140	. . . . . . 7 ⊢ (Fun 𝐹 → ((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
76	75	exlimdv 1966	. . . . . 6 ⊢ (Fun 𝐹 → (∃𝑢(𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
77	15, 66, 76	sylsyld 62	. . . . 5 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
78	77	ralrimivw 3164	. . . 4 ⊢ (𝐹:On–onto→V → ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
79	78	ralrimivw 3164	. . 3 ⊢ (𝐹:On–onto→V → ∀𝑣 ∈ V ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
80		dff13 7259	. . 3 ⊢ (𝐻:V–1-1→On ↔ (𝐻:V⟶On ∧ ∀𝑣 ∈ V ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)))
81	14, 79, 80	sylanbrc 595	. 2 ⊢ (𝐹:On–onto→V → 𝐻:V–1-1→On)
82		onvfowev.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐻‘𝑥) ∈ (𝐻‘𝑦)}
83	82	vonf1wev 35616	. 2 ⊢ (𝐻:V–1-1→On → 𝑅 We V)
84	81, 83	syl 18	1 ⊢ (𝐹:On–onto→V → 𝑅 We V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃!weu 2599   ≠ wne 2961  ∀wral 3082  Vcvv 3458   ⊆ wss 3908  ∅c0 4289  {csn 4594  ∩ cint 4917   class class class wbr 5114  {copab 5178   ↦ cmpt 5197   We wwe 5618  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   “ cima 5669  Oncon0 6367  Fun wfun 6537  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-fv 6551

This theorem is used by: (None)