Theorem onvfowev 36050

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⟶onto𝑋), 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⟶ontoV → 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 6211	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
2		fofn 6820	. . . . . . . 8 ⊢ (𝐹:On⟶ontoV → 𝐹 Fn On)
3	2	fndmd 6666	. . . . . . 7 ⊢ (𝐹:On⟶ontoV → dom 𝐹 = On)
4	1, 3	sseqtrid 3955	. . . . . 6 ⊢ (𝐹:On⟶ontoV → (◡𝐹 “ {𝑧}) ⊆ On)
5		vex 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
6		forn 6821	. . . . . . . 8 ⊢ (𝐹:On⟶ontoV → ran 𝐹 = V)
7	5, 6	eleqtrrid 2871	. . . . . . 7 ⊢ (𝐹:On⟶ontoV → 𝑧 ∈ ran 𝐹)
8		inisegn0 6108	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑧}) ≠ ∅)
9	7, 8	sylib 221	. . . . . 6 ⊢ (𝐹:On⟶ontoV → (◡𝐹 “ {𝑧}) ≠ ∅)
10		oninton 7833	. . . . . 6 ⊢ (((◡𝐹 “ {𝑧}) ⊆ On ∧ (◡𝐹 “ {𝑧}) ≠ ∅) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
11	4, 9, 10	syl2anc 596	. . . . 5 ⊢ (𝐹:On⟶ontoV → ∩ (◡𝐹 “ {𝑧}) ∈ On)
12	11	adantr 486	. . . 4 ⊢ ((𝐹:On⟶ontoV ∧ 𝑧 ∈ V) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
13		onvfowev.2	. . . 4 ⊢ 𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧}))
14	12, 13	fmptd 7138	. . 3 ⊢ (𝐹:On⟶ontoV → 𝐻:V⟶On)
15		fofun 6819	. . . . . 6 ⊢ (𝐹:On⟶ontoV → Fun 𝐹)
16		fvexd 6923	. . . . . . . . 9 ⊢ ((𝐹:On⟶ontoV ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → (𝐻‘𝑣) ∈ V)
17		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐹:On⟶ontoV → 𝑤 ∈ V)
19	11	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On⟶ontoV ∧ 𝑧 = 𝑤) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
20		sneq 4594	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
21	20	imaeq2d 6064	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (◡𝐹 “ {𝑧}) = (◡𝐹 “ {𝑤}))
22	21	inteqd 4915	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑤}))
23	22	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On⟶ontoV ∧ 𝑧 = 𝑤) → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑤}))
24	18, 19, 23	fvmptdv2 7035	. . . . . . . . . . . . . 14 ⊢ (𝐹:On⟶ontoV → (𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧})) → (𝐻‘𝑤) = ∩ (◡𝐹 “ {𝑤})))
25	13, 24	mpi 21	. . . . . . . . . . . . 13 ⊢ (𝐹:On⟶ontoV → (𝐻‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
26		cnvimass 6211	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
27	26, 3	sseqtrid 3955	. . . . . . . . . . . . . 14 ⊢ (𝐹:On⟶ontoV → (◡𝐹 “ {𝑤}) ⊆ On)
28	17, 6	eleqtrrid 2871	. . . . . . . . . . . . . . 15 ⊢ (𝐹:On⟶ontoV → 𝑤 ∈ ran 𝐹)
29		inisegn0 6108	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
30	28, 29	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝐹:On⟶ontoV → (◡𝐹 “ {𝑤}) ≠ ∅)
31		onint 7828	. . . . . . . . . . . . . 14 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
32	27, 30, 31	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝐹:On⟶ontoV → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
33	25, 32	eqeltrd 2864	. . . . . . . . . . . 12 ⊢ (𝐹:On⟶ontoV → (𝐻‘𝑤) ∈ (◡𝐹 “ {𝑤}))
34		eleq1 2852	. . . . . . . . . . . 12 ⊢ ((𝐻‘𝑣) = (𝐻‘𝑤) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤}) ↔ (𝐻‘𝑤) ∈ (◡𝐹 “ {𝑤})))
35	33, 34	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝐹:On⟶ontoV → ((𝐻‘𝑣) = (𝐻‘𝑤) → (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
36		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹:On⟶ontoV → 𝑣 ∈ V)
38	11	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹:On⟶ontoV ∧ 𝑧 = 𝑣) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
39		sneq 4594	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → {𝑧} = {𝑣})
40	39	imaeq2d 6064	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → (◡𝐹 “ {𝑧}) = (◡𝐹 “ {𝑣}))
41	40	inteqd 4915	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑣 → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑣}))
42	41	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐹:On⟶ontoV ∧ 𝑧 = 𝑣) → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑣}))
43	37, 38, 42	fvmptdv2 7035	. . . . . . . . . . . . 13 ⊢ (𝐹:On⟶ontoV → (𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧})) → (𝐻‘𝑣) = ∩ (◡𝐹 “ {𝑣})))
44	13, 43	mpi 21	. . . . . . . . . . . 12 ⊢ (𝐹:On⟶ontoV → (𝐻‘𝑣) = ∩ (◡𝐹 “ {𝑣}))
45		cnvimass 6211	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
46	45, 3	sseqtrid 3955	. . . . . . . . . . . . 13 ⊢ (𝐹:On⟶ontoV → (◡𝐹 “ {𝑣}) ⊆ On)
47	36, 6	eleqtrrid 2871	. . . . . . . . . . . . . 14 ⊢ (𝐹:On⟶ontoV → 𝑣 ∈ ran 𝐹)
48		inisegn0 6108	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
49	47, 48	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝐹:On⟶ontoV → (◡𝐹 “ {𝑣}) ≠ ∅)
50		onint 7828	. . . . . . . . . . . . 13 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
51	46, 49, 50	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝐹:On⟶ontoV → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
52	44, 51	eqeltrd 2864	. . . . . . . . . . 11 ⊢ (𝐹:On⟶ontoV → (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}))
53	35, 52	jctild 535	. . . . . . . . . 10 ⊢ (𝐹:On⟶ontoV → ((𝐻‘𝑣) = (𝐻‘𝑤) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤}))))
54	53	imp 412	. . . . . . . . 9 ⊢ ((𝐹:On⟶ontoV ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
55		eleq1 2852	. . . . . . . . . 10 ⊢ (𝑢 = (𝐻‘𝑣) → (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣})))
56		eleq1 2852	. . . . . . . . . 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 3556	. . . . . . . 8 ⊢ ((𝐹:On⟶ontoV ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → ∃𝑢(𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤})))
59	58	ex 418	. . . . . . 7 ⊢ (𝐹:On⟶ontoV → ((𝐻‘𝑣) = (𝐻‘𝑤) → ∃𝑢(𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤}))))
60		elinisegg 6103	. . . . . . . . . 10 ⊢ ((𝑣 ∈ V ∧ 𝑢 ∈ V) → (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ 𝑢𝐹𝑣))
61	60	el2v 3461	. . . . . . . . 9 ⊢ (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ 𝑢𝐹𝑣)
62		elinisegg 6103	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑢 ∈ V) → (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ 𝑢𝐹𝑤))
63	62	el2v 3461	. . . . . . . . 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⟶ontoV → ((𝐻‘𝑣) = (𝐻‘𝑤) → ∃𝑢(𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤)))
67		funeu 6585	. . . . . . . . . 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 5110	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝑢𝐹𝑣 ↔ 𝑢𝐹𝑤))
71	70	eu4 2643	. . . . . . . . . . 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 2230	. . . . . . . . 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 1968	. . . . . 6 ⊢ (Fun 𝐹 → (∃𝑢(𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
77	15, 66, 76	sylsyld 62	. . . . 5 ⊢ (𝐹:On⟶ontoV → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
78	77	ralrimivw 3162	. . . 4 ⊢ (𝐹:On⟶ontoV → ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
79	78	ralrimivw 3162	. . 3 ⊢ (𝐹:On⟶ontoV → ∀𝑣 ∈ V ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
80		dff13 7281	. . 3 ⊢ (𝐻:V⟶1-1On ↔ (𝐻:V⟶On ∧ ∀𝑣 ∈ V ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)))
81	14, 79, 80	sylanbrc 595	. 2 ⊢ (𝐹:On⟶ontoV → 𝐻:V⟶1-1On)
82		onvfowev.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐻‘𝑥) ∈ (𝐻‘𝑦)}
83	82	vonf1wev 36036	. 2 ⊢ (𝐻:V⟶1-1On → 𝑅 We V)
84	81, 83	syl 18	1 ⊢ (𝐹:On⟶ontoV → 𝑅 We V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588  ∩ cint 4910   class class class wbr 5106  {copab 5170   ↦ cmpt 5189   We wwe 5607  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  Oncon0 6380  Fun wfun 6552  ⟶wf 6554  ⟶1-1wf1 6555  ⟶ontowfo 6556  ‘cfv 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-fv 6566

This theorem is used by: (None)