Theorem vonf1onprcf1ac 36085

Description: If 𝐹 maps the universe one-to-one into the ordinals and 𝐴 is a proper class, then 𝐼 maps the ordinals one-to-one into 𝐴 and the Axiom of Choice holds. This is the ZFC version of (6 → 7) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first hypothesis would be something like 𝜑 → ∀𝑋∃𝐹𝐹:𝑋⟶1-1On, but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 22-Jun-2026.)

Hypotheses

Ref	Expression
vonf1onprcf1ac.1	⊢ (𝜑 → 𝐹:V⟶1-1On)
vonf1onprcf1ac.2	⊢ (𝜑 → ¬ 𝐴 ∈ V)
vonf1onprcf1ac.3	⊢ 𝐼 = (◡(𝐹 ↾ 𝐴) ∘ 𝐻)
vonf1onprcf1ac.4	⊢ 𝐻 = OrdIso( E , (𝐹 “ 𝐴))

Assertion

Ref	Expression
vonf1onprcf1ac	⊢ (𝜑 → (𝐼:On⟶1-1𝐴 ∧ CHOICE))

Proof of Theorem vonf1onprcf1ac

Dummy variables 𝑧 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vonf1onprcf1ac.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:V⟶1-1On)
2		ssv 3937	. . . . . . 7 ⊢ 𝐴 ⊆ V
3		f1ores 6869	. . . . . . 7 ⊢ ((𝐹:V⟶1-1On ∧ 𝐴 ⊆ V) → (𝐹 ↾ 𝐴):𝐴⟶1-1onto(𝐹 “ 𝐴))
4	1, 2, 3	sylancl 598	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶1-1onto(𝐹 “ 𝐴))
5		f1ocnv 6867	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴⟶1-1onto(𝐹 “ 𝐴) → ◡(𝐹 ↾ 𝐴):(𝐹 “ 𝐴)⟶1-1onto𝐴)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → ◡(𝐹 ↾ 𝐴):(𝐹 “ 𝐴)⟶1-1onto𝐴)
7		f1f 6808	. . . . . . . . 9 ⊢ (𝐹:V⟶1-1On → 𝐹:V⟶On)
8	1, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:V⟶On)
9	8	fimassd 6761	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ On)
10		vonf1onprcf1ac.2	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ V)
11		f1preimaex 35826	. . . . . . . . . . 11 ⊢ ((𝐹:V⟶1-1On ∧ 𝐴 ⊆ V ∧ (𝐹 “ 𝐴) ∈ V) → 𝐴 ∈ V)
12	2, 11	mp3an2 1479	. . . . . . . . . 10 ⊢ ((𝐹:V⟶1-1On ∧ (𝐹 “ 𝐴) ∈ V) → 𝐴 ∈ V)
13	12	ex 418	. . . . . . . . 9 ⊢ (𝐹:V⟶1-1On → ((𝐹 “ 𝐴) ∈ V → 𝐴 ∈ V))
14	1, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ 𝐴) ∈ V → 𝐴 ∈ V))
15	10, 14	mtod 201	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐹 “ 𝐴) ∈ V)
16		epweon 7821	. . . . . . . . 9 ⊢ E We On
17		wess 5644	. . . . . . . . 9 ⊢ ((𝐹 “ 𝐴) ⊆ On → ( E We On → E We (𝐹 “ 𝐴)))
18	16, 17	mpi 21	. . . . . . . 8 ⊢ ((𝐹 “ 𝐴) ⊆ On → E We (𝐹 “ 𝐴))
19		epse 5640	. . . . . . . . 9 ⊢ E Se (𝐹 “ 𝐴)
20		vonf1onprcf1ac.4	. . . . . . . . . 10 ⊢ 𝐻 = OrdIso( E , (𝐹 “ 𝐴))
21	20	ordtypeon 35843	. . . . . . . . 9 ⊢ (( E We (𝐹 “ 𝐴) ∧ E Se (𝐹 “ 𝐴) ∧ ¬ (𝐹 “ 𝐴) ∈ V) → 𝐻 Isom E , E (On, (𝐹 “ 𝐴)))
22	19, 21	mp3an2 1479	. . . . . . . 8 ⊢ (( E We (𝐹 “ 𝐴) ∧ ¬ (𝐹 “ 𝐴) ∈ V) → 𝐻 Isom E , E (On, (𝐹 “ 𝐴)))
23	18, 22	sylan 592	. . . . . . 7 ⊢ (((𝐹 “ 𝐴) ⊆ On ∧ ¬ (𝐹 “ 𝐴) ∈ V) → 𝐻 Isom E , E (On, (𝐹 “ 𝐴)))
24	9, 15, 23	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐻 Isom E , E (On, (𝐹 “ 𝐴)))
25		isof1o 7362	. . . . . 6 ⊢ (𝐻 Isom E , E (On, (𝐹 “ 𝐴)) → 𝐻:On⟶1-1onto(𝐹 “ 𝐴))
26	24, 25	syl 18	. . . . 5 ⊢ (𝜑 → 𝐻:On⟶1-1onto(𝐹 “ 𝐴))
27		f1oco 6878	. . . . 5 ⊢ ((◡(𝐹 ↾ 𝐴):(𝐹 “ 𝐴)⟶1-1onto𝐴 ∧ 𝐻:On⟶1-1onto(𝐹 “ 𝐴)) → (◡(𝐹 ↾ 𝐴) ∘ 𝐻):On⟶1-1onto𝐴)
28	6, 26, 27	syl2anc 596	. . . 4 ⊢ (𝜑 → (◡(𝐹 ↾ 𝐴) ∘ 𝐻):On⟶1-1onto𝐴)
29		vonf1onprcf1ac.3	. . . . . 6 ⊢ 𝐼 = (◡(𝐹 ↾ 𝐴) ∘ 𝐻)
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = (◡(𝐹 ↾ 𝐴) ∘ 𝐻))
31	30	f1oeq1d 6849	. . . 4 ⊢ (𝜑 → (𝐼:On⟶1-1onto𝐴 ↔ (◡(𝐹 ↾ 𝐴) ∘ 𝐻):On⟶1-1onto𝐴))
32	28, 31	mpbird 260	. . 3 ⊢ (𝜑 → 𝐼:On⟶1-1onto𝐴)
33		f1of1 6853	. . 3 ⊢ (𝐼:On⟶1-1onto𝐴 → 𝐼:On⟶1-1𝐴)
34	32, 33	syl 18	. 2 ⊢ (𝜑 → 𝐼:On⟶1-1𝐴)
35		eqid 2763	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}
36	35	vonf1wev 36077	. . . . . 6 ⊢ (𝐹:V⟶1-1On → {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We V)
37		ssv 3937	. . . . . . 7 ⊢ 𝑧 ⊆ V
38		wess 5644	. . . . . . 7 ⊢ (𝑧 ⊆ V → ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We V → {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We 𝑧))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We V → {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We 𝑧)
40	1, 36, 39	3syl 19	. . . . 5 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We 𝑧)
41		weinxp 5744	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We 𝑧 ↔ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) We 𝑧)
42		vex 3458	. . . . . . . . 9 ⊢ 𝑧 ∈ V
43	42, 42	xpex 7799	. . . . . . . 8 ⊢ (𝑧 × 𝑧) ∈ V
44	43	inex2 5285	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) ∈ V
45		weeq1 5645	. . . . . . 7 ⊢ (𝑤 = ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) → (𝑤 We 𝑧 ↔ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) We 𝑧))
46	44, 45	spcev 3564	. . . . . 6 ⊢ (({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) We 𝑧 → ∃𝑤 𝑤 We 𝑧)
47	41, 46	sylbi 220	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We 𝑧 → ∃𝑤 𝑤 We 𝑧)
48	40, 47	syl 18	. . . 4 ⊢ (𝜑 → ∃𝑤 𝑤 We 𝑧)
49	48	alrimiv 1962	. . 3 ⊢ (𝜑 → ∀𝑧∃𝑤 𝑤 We 𝑧)
50		dfac8 10253	. . 3 ⊢ (CHOICE ↔ ∀𝑧∃𝑤 𝑤 We 𝑧)
51	49, 50	sylibr 237	. 2 ⊢ (𝜑 → CHOICE)
52	34, 51	jca 521	1 ⊢ (𝜑 → (𝐼:On⟶1-1𝐴 ∧ CHOICE))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  {copab 5173   E cep 5557   Se wse 5609   We wwe 5610   × cxp 5656  ^◡ccnv 5657   ↾ cres 5660   “ cima 5661   ∘ ccom 5662  Oncon0 6388  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567  OrdIsocoi 9540  CHOICEwac 10233

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-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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-rmo 3369  df-reu 3370  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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-en 9012  df-oi 9541  df-card 10057  df-ac 10234

This theorem is used by: (None)