Theorem vonf1oonfo 35867

Description: If 𝐹 is a bijection from the ordinals to the universe and 𝐴 is non-empty, then 𝐻 maps the ordinals onto 𝐴. This is the ZFC version of (5 → 8) in https://tinyurl.com/hamkins-gblac, though it neglects to specify that 𝐴 must be non-empty. Note that in NBG set theory the antecedent would be something like ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:𝑋–1-1-onto→On), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. This theorem can also be viewed as (2 → 8). (Contributed by BTernaryTau, 11-Jun-2026.)

Hypotheses

Ref	Expression
vonf1oonfo.1	⊢ 𝐻 = (𝑥 ∈ On ↦ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
vonf1oonfo.2	⊢ 𝐷 = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})

Assertion

Ref	Expression
vonf1oonfo	⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐻:On–onto→𝐴)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑥,𝐹   𝑦,𝐹

Allowed substitution hints:   𝐷(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem vonf1oonfo

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

Step	Hyp	Ref	Expression
1		vonf1oonfo.1	. . . . 5 ⊢ 𝐻 = (𝑥 ∈ On ↦ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
2	1	rnmpt 5939	. . . 4 ⊢ ran 𝐻 = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)}
3		iffalse 4491	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
4	3	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
5		n0 4300	. . . . . . . . . . . . 13 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐴)
6		19.42v 1986	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) ↔ (𝐹:On–1-1-onto→V ∧ ∃𝑤 𝑤 ∈ 𝐴))
7		f1ofo 6824	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On–1-1-onto→V → 𝐹:On–onto→V)
8		foelcdmi 6938	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:On–onto→V ∧ 𝑤 ∈ V) → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
9	8	elvd 3457	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On–onto→V → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
10	7, 9	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:On–1-1-onto→V → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
11		r19.41v 3193	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦 ∈ On ((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴))
12		eleq1 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
13	12	biimpar 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐴)
14	13	reximi 3101	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦 ∈ On ((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
15	11, 14	sylbir 238	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
16	10, 15	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
17	16	exlimiv 1963	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
18	6, 17	sylbir 238	. . . . . . . . . . . . 13 ⊢ ((𝐹:On–1-1-onto→V ∧ ∃𝑤 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
19	5, 18	sylan2b 606	. . . . . . . . . . . 12 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
20		vonf1oonfo.2	. . . . . . . . . . . . 13 ⊢ 𝐷 = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})
21		nfcv 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝐹
22		nfrab1 3432	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
23	22	nfint 4917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
24	21, 23	nffv 6887	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})
25	24	nfel1 2939	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴
26		fveq2 6877	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴} → (𝐹‘𝑦) = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}))
27	26	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴} → ((𝐹‘𝑦) ∈ 𝐴 ↔ (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴))
28	25, 27	onminsb 7797	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴 → (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴)
29	20, 28	eqeltrid 2865	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴 → 𝐷 ∈ 𝐴)
30	19, 29	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐷 ∈ 𝐴)
31	30	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → 𝐷 ∈ 𝐴)
32	4, 31	eqeltrd 2861	. . . . . . . . 9 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
33	32	3expia 1139	. . . . . . . 8 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴))
34		iftrue 4488	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = (𝐹‘𝑥))
35		id 23	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐴)
36	34, 35	eqeltrd 2861	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
37	33, 36	pm2.61d2 183	. . . . . . 7 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
38		eleq1 2849	. . . . . . 7 ⊢ (𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → (𝑧 ∈ 𝐴 ↔ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴))
39	37, 38	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
40	39	rexlimdvw 3169	. . . . 5 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
41	40	abssdv 4015	. . . 4 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)} ⊆ 𝐴)
42	2, 41	eqsstrid 3969	. . 3 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ran 𝐻 ⊆ 𝐴)
43		fveqeq2 6886	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑧) → ((𝐹‘𝑥) = 𝑧 ↔ (𝐹‘(◡𝐹‘𝑧)) = 𝑧))
44		f1ocnvdm 7285	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑧 ∈ V) → (◡𝐹‘𝑧) ∈ On)
45	44	elvd 3457	. . . . . . . . 9 ⊢ (𝐹:On–1-1-onto→V → (◡𝐹‘𝑧) ∈ On)
46		f1ocnvfv2 7277	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑧 ∈ V) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
47	46	elvd 3457	. . . . . . . . 9 ⊢ (𝐹:On–1-1-onto→V → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
48	43, 45, 47	rspcedvdw 3580	. . . . . . . 8 ⊢ (𝐹:On–1-1-onto→V → ∃𝑥 ∈ On (𝐹‘𝑥) = 𝑧)
49		eleq1 2849	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑧 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
50	49	biimpar 483	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐴)
51	50	iftrued 4490	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = (𝐹‘𝑥))
52		simpl 488	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑥) = 𝑧)
53	51, 52	eqtr2d 2797	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
54	53	expcom 419	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑧 → 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
55	54	reximdv 3178	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (∃𝑥 ∈ On (𝐹‘𝑥) = 𝑧 → ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
56	48, 55	syl5com 32	. . . . . . 7 ⊢ (𝐹:On–1-1-onto→V → (𝑧 ∈ 𝐴 → ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
57	56	ralrimiv 3154	. . . . . 6 ⊢ (𝐹:On–1-1-onto→V → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
58		ssabral 4012	. . . . . 6 ⊢ (𝐴 ⊆ {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)} ↔ ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
59	57, 58	sylibr 237	. . . . 5 ⊢ (𝐹:On–1-1-onto→V → 𝐴 ⊆ {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)})
60	59, 2	sseqtrrdi 3972	. . . 4 ⊢ (𝐹:On–1-1-onto→V → 𝐴 ⊆ ran 𝐻)
61	60	adantr 486	. . 3 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ ran 𝐻)
62	42, 61	eqssd 3948	. 2 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ran 𝐻 = 𝐴)
63		fvex 6890	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
64	20	fvexi 6891	. . . . 5 ⊢ 𝐷 ∈ V
65	63, 64	ifex 4533	. . . 4 ⊢ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ V
66	65, 1	fnmpti 6674	. . 3 ⊢ 𝐻 Fn On
67		df-fo 6537	. . 3 ⊢ (𝐻:On–onto→𝐴 ↔ (𝐻 Fn On ∧ ran 𝐻 = 𝐴))
68	66, 67	mpbiran 722	. 2 ⊢ (𝐻:On–onto→𝐴 ↔ ran 𝐻 = 𝐴)
69	62, 68	sylibr 237	1 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐻:On–onto→𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  ifcif 4482  ∩ cint 4907   ↦ cmpt 5186  ^◡ccnv 5650  ran crn 5652  Oncon0 6355   Fn wfn 6526  –onto→wfo 6529  –1-1-onto→wf1o 6530  ‘cfv 6531

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  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-ord 6358  df-on 6359  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539

This theorem is used by: (None)