Theorem onvfowev 35471

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 6075	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
2		fofn 6784	. . . . . . . 8 ⊢ (𝐹:On–onto→V → 𝐹 Fn On)
3	2	fndmd 6630	. . . . . . 7 ⊢ (𝐹:On–onto→V → dom 𝐹 = On)
4	1, 3	sseqtrid 3981	. . . . . 6 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑧}) ⊆ On)
5		vex 3461	. . . . . . . 8 ⊢ 𝑧 ∈ V
6		forn 6785	. . . . . . . 8 ⊢ (𝐹:On–onto→V → ran 𝐹 = V)
7	5, 6	eleqtrrid 2872	. . . . . . 7 ⊢ (𝐹:On–onto→V → 𝑧 ∈ ran 𝐹)
8		inisegn0 6091	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑧}) ≠ ∅)
9	7, 8	sylib 221	. . . . . 6 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑧}) ≠ ∅)
10		oninton 7782	. . . . . 6 ⊢ (((◡𝐹 “ {𝑧}) ⊆ On ∧ (◡𝐹 “ {𝑧}) ≠ ∅) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
11	4, 9, 10	syl2anc 595	. . . . 5 ⊢ (𝐹:On–onto→V → ∩ (◡𝐹 “ {𝑧}) ∈ On)
12	11	adantr 485	. . . 4 ⊢ ((𝐹:On–onto→V ∧ 𝑧 ∈ V) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
13		onvfowev.2	. . . 4 ⊢ 𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧}))
14	12, 13	fmptd 7099	. . 3 ⊢ (𝐹:On–onto→V → 𝐻:V⟶On)
15		fofun 6783	. . . . . 6 ⊢ (𝐹:On–onto→V → Fun 𝐹)
16		fvexd 6886	. . . . . . . . 9 ⊢ ((𝐹:On–onto→V ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → (𝐻‘𝑣) ∈ V)
17		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐹:On–onto→V → 𝑤 ∈ V)
19	11	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑤) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
20		sneq 4595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
21	20	imaeq2d 6053	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (◡𝐹 “ {𝑧}) = (◡𝐹 “ {𝑤}))
22	21	inteqd 4913	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑤}))
23	22	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑤) → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑤}))
24	18, 19, 23	fvmptdv2 6998	. . . . . . . . . . . . . 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 6075	. . . . . . . . . . . . . . 15 ⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹
27	26, 3	sseqtrid 3981	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑤}) ⊆ On)
28	17, 6	eleqtrrid 2872	. . . . . . . . . . . . . . 15 ⊢ (𝐹:On–onto→V → 𝑤 ∈ ran 𝐹)
29		inisegn0 6091	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅)
30	28, 29	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑤}) ≠ ∅)
31		onint 7777	. . . . . . . . . . . . . 14 ⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
32	27, 30, 31	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}))
33	25, 32	eqeltrd 2865	. . . . . . . . . . . 12 ⊢ (𝐹:On–onto→V → (𝐻‘𝑤) ∈ (◡𝐹 “ {𝑤}))
34		eleq1 2853	. . . . . . . . . . . 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 3461	. . . . . . . . . . . . . . 15 ⊢ 𝑣 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → 𝑣 ∈ V)
38	11	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑣) → ∩ (◡𝐹 “ {𝑧}) ∈ On)
39		sneq 4595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → {𝑧} = {𝑣})
40	39	imaeq2d 6053	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → (◡𝐹 “ {𝑧}) = (◡𝐹 “ {𝑣}))
41	40	inteqd 4913	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑣 → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑣}))
42	41	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹:On–onto→V ∧ 𝑧 = 𝑣) → ∩ (◡𝐹 “ {𝑧}) = ∩ (◡𝐹 “ {𝑣}))
43	37, 38, 42	fvmptdv2 6998	. . . . . . . . . . . . 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 6075	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹
46	45, 3	sseqtrid 3981	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑣}) ⊆ On)
47	36, 6	eleqtrrid 2872	. . . . . . . . . . . . . 14 ⊢ (𝐹:On–onto→V → 𝑣 ∈ ran 𝐹)
48		inisegn0 6091	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅)
49	47, 48	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝐹:On–onto→V → (◡𝐹 “ {𝑣}) ≠ ∅)
50		onint 7777	. . . . . . . . . . . . 13 ⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
51	46, 49, 50	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝐹:On–onto→V → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}))
52	44, 51	eqeltrd 2865	. . . . . . . . . . 11 ⊢ (𝐹:On–onto→V → (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}))
53	35, 52	jctild 534	. . . . . . . . . 10 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤}))))
54	53	imp 411	. . . . . . . . 9 ⊢ ((𝐹:On–onto→V ∧ (𝐻‘𝑣) = (𝐻‘𝑤)) → ((𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣}) ∧ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
55		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑢 = (𝐻‘𝑣) → (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑣})))
56		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑢 = (𝐻‘𝑣) → (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ (𝐻‘𝑣) ∈ (◡𝐹 “ {𝑤})))
57	55, 56	anbi12d 643	. . . . . . . . 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 417	. . . . . . 7 ⊢ (𝐹:On–onto→V → ((𝐻‘𝑣) = (𝐻‘𝑤) → ∃𝑢(𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤}))))
60		elinisegg 6086	. . . . . . . . . 10 ⊢ ((𝑣 ∈ V ∧ 𝑢 ∈ V) → (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ 𝑢𝐹𝑣))
61	60	el2v 3464	. . . . . . . . 9 ⊢ (𝑢 ∈ (◡𝐹 “ {𝑣}) ↔ 𝑢𝐹𝑣)
62		elinisegg 6086	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑢 ∈ V) → (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ 𝑢𝐹𝑤))
63	62	el2v 3464	. . . . . . . . 9 ⊢ (𝑢 ∈ (◡𝐹 “ {𝑤}) ↔ 𝑢𝐹𝑤)
64	61, 63	anbi12i 639	. . . . . . . 8 ⊢ ((𝑢 ∈ (◡𝐹 “ {𝑣}) ∧ 𝑢 ∈ (◡𝐹 “ {𝑤})) ↔ (𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤))
65	64	exbii 1871	. . . . . . 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 6550	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣) → ∃!𝑣 𝑢𝐹𝑣)
68	67	3adant3 1148	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → ∃!𝑣 𝑢𝐹𝑣)
69		3simpc 1166	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → (𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤))
70		breq2 5109	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝑢𝐹𝑣 ↔ 𝑢𝐹𝑤))
71	70	eu4 2645	. . . . . . . . . . 11 ⊢ (∃!𝑣 𝑢𝐹𝑣 ↔ (∃𝑣 𝑢𝐹𝑣 ∧ ∀𝑣∀𝑤((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤)))
72	71	simprbi 502	. . . . . . . . . 10 ⊢ (∃!𝑣 𝑢𝐹𝑣 → ∀𝑣∀𝑤((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
73	72	19.21bbi 2228	. . . . . . . . 9 ⊢ (∃!𝑣 𝑢𝐹𝑣 → ((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
74	68, 69, 73	sylc 66	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤)
75	74	3expib 1138	. . . . . . 7 ⊢ (Fun 𝐹 → ((𝑢𝐹𝑣 ∧ 𝑢𝐹𝑤) → 𝑣 = 𝑤))
76	75	exlimdv 1956	. . . . . 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 3161	. . . 4 ⊢ (𝐹:On–onto→V → ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
79	78	ralrimivw 3161	. . 3 ⊢ (𝐹:On–onto→V → ∀𝑣 ∈ V ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))
80		dff13 7242	. . 3 ⊢ (𝐻:V–1-1→On ↔ (𝐻:V⟶On ∧ ∀𝑣 ∈ V ∀𝑤 ∈ V ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)))
81	14, 79, 80	sylanbrc 594	. 2 ⊢ (𝐹:On–onto→V → 𝐻:V–1-1→On)
82		onvfowev.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐻‘𝑥) ∈ (𝐻‘𝑦)}
83	82	vonf1wev 35463	. 2 ⊢ (𝐻:V–1-1→On → 𝑅 We V)
84	81, 83	syl 18	1 ⊢ (𝐹:On–onto→V → 𝑅 We V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃!weu 2598   ≠ wne 2960  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  {csn 4585  ∩ cint 4908   class class class wbr 5105  {copab 5167   ↦ cmpt 5186   We wwe 5604  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   “ cima 5655  Oncon0 6350  Fun wfun 6519  ⟶wf 6521  –1-1→wf1 6522  –onto→wfo 6523  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-fv 6533

This theorem is referenced by: (None)