Theorem onvf1od 35309

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 8331	. . . 4 ⊢ 𝐹 Fn On
3		dffn2 6666	. . . 4 ⊢ (𝐹 Fn On ↔ 𝐹:On⟶V)
4	2, 3	mpbi 230	. . 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 2737	. . . . . . . 8 ⊢ ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)} = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}
8		eqid 2737	. . . . . . . 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 35307	. . . . . . 7 ⊢ (𝑡 ∈ On → (𝐹‘𝑡) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → (𝐹‘𝑡) = (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))))
11		fnfun 6594	. . . . . . . . . 10 ⊢ (𝐹 Fn On → Fun 𝐹)
12		vex 3434	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
13	12	funimaex 6582	. . . . . . . . . 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 35306	. . . . . . . . 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 3903	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ (𝐹 “ 𝑡))
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ¬ (𝐺‘((𝑅1‘∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}) ∖ (𝐹 “ 𝑡))) ∈ (𝐹 “ 𝑡))
20	10, 19	eqneltrd 2857	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ On) → ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
21	20	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
22		fvex 6849	. . . . . . 7 ⊢ (𝐹‘𝑡) ∈ V
23		eldif 3900	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ((𝐹‘𝑡) ∈ V ∧ ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡)))
24	22, 23	mpbiran 710	. . . . . 6 ⊢ ((𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
25	24	ralbii 3084	. . . . 5 ⊢ (∀𝑡 ∈ On (𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) ↔ ∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡))
26	2	tz7.48-2 8376	. . . . 5 ⊢ (∀𝑡 ∈ On (𝐹‘𝑡) ∈ (V ∖ (𝐹 “ 𝑡)) → Fun ◡𝐹)
27	25, 26	sylbir 235	. . . 4 ⊢ (∀𝑡 ∈ On ¬ (𝐹‘𝑡) ∈ (𝐹 “ 𝑡) → Fun ◡𝐹)
28	21, 27	syl 17	. . 3 ⊢ (𝜑 → Fun ◡𝐹)
29		df-f1 6499	. . . 4 ⊢ (𝐹:On–1-1→V ↔ (𝐹:On⟶V ∧ Fun ◡𝐹))
30	29	biimpri 228	. . 3 ⊢ ((𝐹:On⟶V ∧ Fun ◡𝐹) → 𝐹:On–1-1→V)
31	4, 28, 30	sylancr 588	. 2 ⊢ (𝜑 → 𝐹:On–1-1→V)
32		onprc 7727	. . . 4 ⊢ ¬ On ∈ V
33		f1f1orn 6787	. . . . . . 7 ⊢ (𝐹:On–1-1→V → 𝐹:On–1-1-onto→ran 𝐹)
34		f1of1 6775	. . . . . . 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 7905	. . . . . 6 ⊢ ((𝐹:On–1-1→ran 𝐹 ∧ ran 𝐹 ∈ V) → On ∈ V)
37	35, 36	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ∈ V) → On ∈ V)
38	37	stoic1a 1774	. . . 4 ⊢ ((𝜑 ∧ ¬ On ∈ V) → ¬ ran 𝐹 ∈ V)
39	32, 38	mpan2 692	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ∈ V)
40	15, 5, 6, 1, 7, 8	onvf1odlem4 35308	. . 3 ⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))
41	39, 40	mpd 15	. 2 ⊢ (𝜑 → ran 𝐹 = V)
42		dff1o5 6785	. 2 ⊢ (𝐹:On–1-1-onto→V ↔ (𝐹:On–1-1→V ∧ ran 𝐹 = V))
43	31, 41, 42	sylanbrc 584	1 ⊢ (𝜑 → 𝐹:On–1-1-onto→V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887  ∅c0 4274  ∩ cint 4890   ↦ cmpt 5167  ^◡ccnv 5625  ran crn 5627   “ cima 5629  Oncon0 6319  Fun wfun 6488   Fn wfn 6489  ⟶wf 6490  –1-1→wf1 6491  –1-1-onto→wf1o 6493  ‘cfv 6494  recscrecs 8305  𝑅₁cr1 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-reg 9502  ax-inf2 9557

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-r1 9683  df-rank 9684

This theorem is referenced by: (None)