Theorem onvf1od 35691

Description: If 𝐺 is a global choice function, then 𝐹 is a bijection from the ordinals to the universe. This is the ZFC version of (1 → 2) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 5-Dec-2025.)

Hypotheses

Ref	Expression
onvf1od.1	⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
onvf1od.2	⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}
onvf1od.3	⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))
onvf1od.4	⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))

Assertion

Ref	Expression
onvf1od	⊢ (𝜑 → 𝐹:On–1-1-onto→V)

Distinct variable groups:   𝑧,𝐺   𝑤,𝐺   𝑥,𝑤,𝑦   𝑧,𝐹

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤)   𝐹(𝑥, 𝑦, 𝑤)   𝐺(𝑥, 𝑦)   𝑀(𝑥, 𝑦, 𝑧, 𝑤)   𝑁(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem onvf1od

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

Step	Hyp	Ref	Expression
1		onvf1od.4	. . . . 5 ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))
2	1	tfr1 8389	. . . 4 ⊢ 𝐹 Fn On
3		dffn2 6708	. . . 4 ⊢ (𝐹 Fn On ↔ 𝐹:On⟶V)
4	2, 3	mpbi 233	. . 3 ⊢ 𝐹:On⟶V
5		onvf1od.2	. . . . . . . 8 ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}
6		onvf1od.3	. . . . . . . 8 ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))
7		eqid 2762	. . . . . . . 8 ⊢ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}
8		eqid 2762	. . . . . . . 8 ⊢ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡)))
9	5, 6, 1, 7, 8	onvf1odlem3 35689	. . . . . . 7 ⊢ (𝑡 ∈ On → (𝐹‘𝑡) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
10	9	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
11		fnfun 6636	. . . . . . . . . 10 ⊢ (𝐹 Fn On → Fun 𝐹)
12		vex 3457	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
13	12	funimaex 6624	. . . . . . . . . 10 ⊢ (Fun 𝐹 → (𝐹 “ 𝑡) ∈ V)
14	2, 11, 13	mp2b 10	. . . . . . . . 9 ⊢ (𝐹 “ 𝑡) ∈ V
15		onvf1od.1	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
16	15, 7, 8	onvf1odlem2 35688	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 “ 𝑡) ∈ V → (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ ((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
17	14, 16	mpi 21	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ ((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡)))
18	17	eldifbd 3915	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ (𝐹 “ 𝑡))
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ¬ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ (𝐹 “ 𝑡))
20	10, 19	eqneltrd 2882	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
21	20	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
22		fvex 6895	. . . . . . 7 ⊢ (𝐹‘𝑡) ∈ V
23		eldif 3912	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ((𝐹‘𝑡) ∈ V ∧ ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡)))
24	22, 23	mpbiran 722	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
25	24	ralbii 3110	. . . . 5 ⊢ (∀𝑡 ∈ On (𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
26	2	tz7.48-2 8434	. . . . 5 ⊢ (∀𝑡 ∈ On (𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) → Fun ◡𝐹)
27	25, 26	sylbir 238	. . . 4 ⊢ (∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡) → Fun ◡𝐹)
28	21, 27	syl 18	. . 3 ⊢ (𝜑 → Fun ◡𝐹)
29		df-f1 6542	. . . 4 ⊢ (𝐹:On–1-1→V ↔ (𝐹:On⟶V ∧ Fun ◡𝐹))
30	29	biimpri 231	. . 3 ⊢ ((𝐹:On⟶V ∧ Fun ◡𝐹) → 𝐹:On–1-1→V)
31	4, 28, 30	sylancr 599	. 2 ⊢ (𝜑 → 𝐹:On–1-1→V)
32		onprc 7780	. . . 4 ⊢ ¬ On ∈ V
33		f1f1orn 6833	. . . . . . 7 ⊢ (𝐹:On–1-1→V → 𝐹:On–1-1-onto→ran 𝐹)
34		f1of1 6820	. . . . . . 7 ⊢ (𝐹:On–1-1-onto→ran 𝐹 → 𝐹:On–1-1→ran 𝐹)
35	31, 33, 34	3syl 19	. . . . . 6 ⊢ (𝜑 → 𝐹:On–1-1→ran 𝐹)
36		f1dmex 7957	. . . . . 6 ⊢ ((𝐹:On–1-1→ran 𝐹 ∧ ran 𝐹 ∈ V) → On ∈ V)
37	35, 36	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ∈ V) → On ∈ V)
38	37	stoic1a 1805	. . . 4 ⊢ ((𝜑 ∧ ¬ On ∈ V) → ¬ ran 𝐹 ∈ V)
39	32, 38	mpan2 704	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ∈ V)
40	15, 5, 6, 1, 7, 8	onvf1odlem4 35690	. . 3 ⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))
41	39, 40	mpd 16	. 2 ⊢ (𝜑 → ran 𝐹 = V)
42		dff1o5 6831	. 2 ⊢ (𝐹:On–1-1-onto→V ↔ (𝐹:On–1-1→V ∧ ran 𝐹 = V))
43	31, 41, 42	sylanbrc 595	1 ⊢ (𝜑 → 𝐹:On–1-1-onto→V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ∖ cdif 3899  ∅c0 4282  ∩ cint 4910   ↦ cmpt 5190  ^◡ccnv 5658  ran crn 5660   “ cima 5662  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  recscrecs 8362  𝑅₁cr1 9747

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-reg 9567  ax-inf2 9623

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9749  df-rank 9750

This theorem is used by: (None)