Theorem vonf1oonfo 35470

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 5938	. . . 4 ⊢ ran 𝐻 = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)}
3		iffalse 4492	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
4	3	3ad2ant3 1151	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
5		n0 4308	. . . . . . . . . . . . 13 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐴)
6		19.42v 1976	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) ↔ (𝐹:On–1-1-onto→V ∧ ∃𝑤 𝑤 ∈ 𝐴))
7		f1ofo 6818	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On–1-1-onto→V → 𝐹:On–onto→V)
8		foelcdmi 6932	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:On–onto→V ∧ 𝑤 ∈ V) → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
9	8	elvd 3463	. . . . . . . . . . . . . . . . 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 3195	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦 ∈ On ((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴))
12		eleq1 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
13	12	biimpar 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐴)
14	13	reximi 3103	. . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
17	16	exlimiv 1953	. . . . . . . . . . . . . 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 605	. . . . . . . . . . . 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 2927	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝐹
22		nfrab1 3437	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
23	22	nfint 4918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
24	21, 23	nffv 6881	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})
25	24	nfel1 2943	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴
26		fveq2 6871	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴} → (𝐹‘𝑦) = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}))
27	26	eleq1d 2850	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴} → ((𝐹‘𝑦) ∈ 𝐴 ↔ (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴))
28	25, 27	onminsb 7781	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴 → (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴)
29	20, 28	eqeltrid 2869	. . . . . . . . . . . 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 1148	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → 𝐷 ∈ 𝐴)
32	4, 31	eqeltrd 2865	. . . . . . . . 9 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
33	32	3expia 1137	. . . . . . . 8 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴))
34		iftrue 4489	. . . . . . . . 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 2865	. . . . . . . 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 2853	. . . . . . 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 3171	. . . . 5 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
41	40	abssdv 4023	. . . 4 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)} ⊆ 𝐴)
42	2, 41	eqsstrid 3977	. . 3 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ran 𝐻 ⊆ 𝐴)
43		fveqeq2 6880	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑧) → ((𝐹‘𝑥) = 𝑧 ↔ (𝐹‘(◡𝐹‘𝑧)) = 𝑧))
44		f1ocnvdm 7273	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑧 ∈ V) → (◡𝐹‘𝑧) ∈ On)
45	44	elvd 3463	. . . . . . . . 9 ⊢ (𝐹:On–1-1-onto→V → (◡𝐹‘𝑧) ∈ On)
46		f1ocnvfv2 7265	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑧 ∈ V) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
47	46	elvd 3463	. . . . . . . . 9 ⊢ (𝐹:On–1-1-onto→V → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
48	43, 45, 47	rspcedvdw 3587	. . . . . . . 8 ⊢ (𝐹:On–1-1-onto→V → ∃𝑥 ∈ On (𝐹‘𝑥) = 𝑧)
49		eleq1 2853	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = 𝑧 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
50	49	biimpar 482	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐴)
51	50	iftrued 4491	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = (𝐹‘𝑥))
52		simpl 487	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑥) = 𝑧)
53	51, 52	eqtr2d 2801	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
54	53	expcom 418	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑧 → 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
55	54	reximdv 3180	. . . . . . . 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 3156	. . . . . 6 ⊢ (𝐹:On–1-1-onto→V → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
58		ssabral 4020	. . . . . 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 3980	. . . 4 ⊢ (𝐹:On–1-1-onto→V → 𝐴 ⊆ ran 𝐻)
61	60	adantr 485	. . 3 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ ran 𝐻)
62	42, 61	eqssd 3956	. 2 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ran 𝐻 = 𝐴)
63		fvex 6884	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
64	20	fvexi 6885	. . . . 5 ⊢ 𝐷 ∈ V
65	63, 64	ifex 4534	. . . 4 ⊢ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ V
66	65, 1	fnmpti 6668	. . 3 ⊢ 𝐻 Fn On
67		df-fo 6531	. . 3 ⊢ (𝐻:On–onto→𝐴 ↔ (𝐻 Fn On ∧ ran 𝐻 = 𝐴))
68	66, 67	mpbiran 721	. 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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  ifcif 4483  ∩ cint 4908   ↦ cmpt 5186  ^◡ccnv 5651  ran crn 5653  Oncon0 6350   Fn wfn 6520  –onto→wfo 6523  –1-1-onto→wf1o 6524  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533

This theorem is referenced by: (None)