Theorem vonf1onprcf1ac 35995

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–1-1→On, 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–1-1→On)
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–1-1→𝐴 ∧ 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–1-1→On)
2		ssv 3933	. . . . . . 7 ⊢ 𝐴 ⊆ V
3		f1ores 6844	. . . . . . 7 ⊢ ((𝐹:V–1-1→On ∧ 𝐴 ⊆ V) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
4	1, 2, 3	sylancl 598	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
5		f1ocnv 6842	. . . . . 6 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) → ◡(𝐹 ↾ 𝐴):(𝐹 “ 𝐴)–1-1-onto→𝐴)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → ◡(𝐹 ↾ 𝐴):(𝐹 “ 𝐴)–1-1-onto→𝐴)
7		f1f 6783	. . . . . . . . 9 ⊢ (𝐹:V–1-1→On → 𝐹:V⟶On)
8	1, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:V⟶On)
9	8	fimassd 6736	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ On)
10		vonf1onprcf1ac.2	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ V)
11		f1preimaex 35765	. . . . . . . . . . 11 ⊢ ((𝐹:V–1-1→On ∧ 𝐴 ⊆ V ∧ (𝐹 “ 𝐴) ∈ V) → 𝐴 ∈ V)
12	2, 11	mp3an2 1478	. . . . . . . . . 10 ⊢ ((𝐹:V–1-1→On ∧ (𝐹 “ 𝐴) ∈ V) → 𝐴 ∈ V)
13	12	ex 418	. . . . . . . . 9 ⊢ (𝐹:V–1-1→On → ((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 7793	. . . . . . . . 9 ⊢ E We On
17		wess 5637	. . . . . . . . 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 5633	. . . . . . . . 9 ⊢ E Se (𝐹 “ 𝐴)
20		vonf1onprcf1ac.4	. . . . . . . . . 10 ⊢ 𝐻 = OrdIso( E , (𝐹 “ 𝐴))
21	20	ordtypeon 35776	. . . . . . . . 9 ⊢ (( E We (𝐹 “ 𝐴) ∧ E Se (𝐹 “ 𝐴) ∧ ¬ (𝐹 “ 𝐴) ∈ V) → 𝐻 Isom E , E (On, (𝐹 “ 𝐴)))
22	19, 21	mp3an2 1478	. . . . . . . 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 7335	. . . . . 6 ⊢ (𝐻 Isom E , E (On, (𝐹 “ 𝐴)) → 𝐻:On–1-1-onto→(𝐹 “ 𝐴))
26	24, 25	syl 18	. . . . 5 ⊢ (𝜑 → 𝐻:On–1-1-onto→(𝐹 “ 𝐴))
27		f1oco 6853	. . . . 5 ⊢ ((◡(𝐹 ↾ 𝐴):(𝐹 “ 𝐴)–1-1-onto→𝐴 ∧ 𝐻:On–1-1-onto→(𝐹 “ 𝐴)) → (◡(𝐹 ↾ 𝐴) ∘ 𝐻):On–1-1-onto→𝐴)
28	6, 26, 27	syl2anc 596	. . . 4 ⊢ (𝜑 → (◡(𝐹 ↾ 𝐴) ∘ 𝐻):On–1-1-onto→𝐴)
29		vonf1onprcf1ac.3	. . . . . 6 ⊢ 𝐼 = (◡(𝐹 ↾ 𝐴) ∘ 𝐻)
30	29	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = (◡(𝐹 ↾ 𝐴) ∘ 𝐻))
31	30	f1oeq1d 6824	. . . 4 ⊢ (𝜑 → (𝐼:On–1-1-onto→𝐴 ↔ (◡(𝐹 ↾ 𝐴) ∘ 𝐻):On–1-1-onto→𝐴))
32	28, 31	mpbird 260	. . 3 ⊢ (𝜑 → 𝐼:On–1-1-onto→𝐴)
33		f1of1 6828	. . 3 ⊢ (𝐼:On–1-1-onto→𝐴 → 𝐼:On–1-1→𝐴)
34	32, 33	syl 18	. 2 ⊢ (𝜑 → 𝐼:On–1-1→𝐴)
35		eqid 2760	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}
36	35	vonf1wev 35987	. . . . . 6 ⊢ (𝐹:V–1-1→On → {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We V)
37		ssv 3933	. . . . . . 7 ⊢ 𝑧 ⊆ V
38		wess 5637	. . . . . . 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 5737	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} We 𝑧 ↔ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) We 𝑧)
42		vex 3454	. . . . . . . . 9 ⊢ 𝑧 ∈ V
43	42, 42	xpex 7771	. . . . . . . 8 ⊢ (𝑧 × 𝑧) ∈ V
44	43	inex2 5278	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ∩ (𝑧 × 𝑧)) ∈ V
45		weeq1 5638	. . . . . . 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 3560	. . . . . 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 1959	. . 3 ⊢ (𝜑 → ∀𝑧∃𝑤 𝑤 We 𝑧)
50		dfac8 10221	. . 3 ⊢ (CHOICE ↔ ∀𝑧∃𝑤 𝑤 We 𝑧)
51	49, 50	sylibr 237	. 2 ⊢ (𝜑 → CHOICE)
52	34, 51	jca 521	1 ⊢ (𝜑 → (𝐼:On–1-1→𝐴 ∧ CHOICE))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  {copab 5166   E cep 5550   Se wse 5602   We wwe 5603   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Oncon0 6365  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544  OrdIsocoi 9510  CHOICEwac 10201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-en 8982  df-oi 9511  df-card 10027  df-ac 10202

This theorem is used by: (None)