Theorem onvf1od 35454

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 8370	. . . 4 ⊢ 𝐹 Fn On
3		dffn2 6695	. . . 4 ⊢ (𝐹 Fn On ↔ 𝐹:On⟶V)
4	2, 3	mpbi 232	. . 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 2764	. . . . . . . 8 ⊢ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}
8		eqid 2764	. . . . . . . 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 35452	. . . . . . 7 ⊢ (𝑡 ∈ On → (𝐹‘𝑡) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
10	9	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
11		fnfun 6623	. . . . . . . . . 10 ⊢ (𝐹 Fn On → Fun 𝐹)
12		vex 3460	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
13	12	funimaex 6611	. . . . . . . . . 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 35451	. . . . . . . . 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 20	. . . . . . . 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 3919	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ (𝐹 “ 𝑡))
19	18	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ¬ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ (𝐹 “ 𝑡))
20	10, 19	eqneltrd 2884	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
21	20	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
22		fvex 6882	. . . . . . 7 ⊢ (𝐹‘𝑡) ∈ V
23		eldif 3916	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ((𝐹‘𝑡) ∈ V ∧ ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡)))
24	22, 23	mpbiran 719	. . . . . 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 8415	. . . . 5 ⊢ (∀𝑡 ∈ On (𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) → Fun ◡𝐹)
27	25, 26	sylbir 237	. . . 4 ⊢ (∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡) → Fun ◡𝐹)
28	21, 27	syl 17	. . 3 ⊢ (𝜑 → Fun ◡𝐹)
29		df-f1 6528	. . . 4 ⊢ (𝐹:On–1-1→V ↔ (𝐹:On⟶V ∧ Fun ◡𝐹))
30	29	biimpri 230	. . 3 ⊢ ((𝐹:On⟶V ∧ Fun ◡𝐹) → 𝐹:On–1-1→V)
31	4, 28, 30	sylancr 596	. 2 ⊢ (𝜑 → 𝐹:On–1-1→V)
32		onprc 7763	. . . 4 ⊢ ¬ On ∈ V
33		f1f1orn 6820	. . . . . . 7 ⊢ (𝐹:On–1-1→V → 𝐹:On–1-1-onto→ran 𝐹)
34		f1of1 6807	. . . . . . 7 ⊢ (𝐹:On–1-1-onto→ran 𝐹 → 𝐹:On–1-1→ran 𝐹)
35	31, 33, 34	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:On–1-1→ran 𝐹)
36		f1dmex 7940	. . . . . 6 ⊢ ((𝐹:On–1-1→ran 𝐹 ∧ ran 𝐹 ∈ V) → On ∈ V)
37	35, 36	sylan 589	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ∈ V) → On ∈ V)
38	37	stoic1a 1794	. . . 4 ⊢ ((𝜑 ∧ ¬ On ∈ V) → ¬ ran 𝐹 ∈ V)
39	32, 38	mpan2 701	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ∈ V)
40	15, 5, 6, 1, 7, 8	onvf1odlem4 35453	. . 3 ⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))
41	39, 40	mpd 15	. 2 ⊢ (𝜑 → ran 𝐹 = V)
42		dff1o5 6818	. 2 ⊢ (𝐹:On–1-1-onto→V ↔ (𝐹:On–1-1→V ∧ ran 𝐹 = V))
43	31, 41, 42	sylanbrc 592	1 ⊢ (𝜑 → 𝐹:On–1-1-onto→V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088  {crab 3416  Vcvv 3456   ∖ cdif 3903  ∅c0 4287  ∩ cint 4907   ↦ cmpt 5183  ^◡ccnv 5648  ran crn 5650   “ cima 5652  Oncon0 6348  Fun wfun 6517   Fn wfn 6518  ⟶wf 6519  –1-1→wf1 6520  –1-1-onto→wf1o 6522  ‘cfv 6523  recscrecs 8343  𝑅₁cr1 9722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-r1 9724  df-rank 9725

This theorem is referenced by: (None)