Theorem vonf1oonfo 36049

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⟶1-1ontoOn), 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⟶1-1ontoV ∧ 𝐴 ≠ ∅) → 𝐻:On⟶onto𝐴)

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 5946	. . . 4 ⊢ ran 𝐻 = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)}
3		iffalse 4494	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
4	3	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
5		n0 4303	. . . . . . . . . . . . 13 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐴)
6		19.42v 1988	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On⟶1-1ontoV ∧ 𝑤 ∈ 𝐴) ↔ (𝐹:On⟶1-1ontoV ∧ ∃𝑤 𝑤 ∈ 𝐴))
7		f1ofo 6854	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On⟶1-1ontoV → 𝐹:On⟶ontoV)
8		foelcdmi 6969	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:On⟶ontoV ∧ 𝑤 ∈ V) → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
9	8	elvd 3460	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On⟶ontoV → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
10	7, 9	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:On⟶1-1ontoV → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
11		r19.41v 3196	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦 ∈ On ((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴))
12		eleq1 2852	. . . . . . . . . . . . . . . . . . 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 3104	. . . . . . . . . . . . . . . . 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⟶1-1ontoV ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
17	16	exlimiv 1965	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On⟶1-1ontoV ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
18	6, 17	sylbir 238	. . . . . . . . . . . . 13 ⊢ ((𝐹:On⟶1-1ontoV ∧ ∃𝑤 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
19	5, 18	sylan2b 606	. . . . . . . . . . . 12 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
20		vonf1oonfo.2	. . . . . . . . . . . . 13 ⊢ 𝐷 = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})
21		nfcv 2926	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝐹
22		nfrab1 3435	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
23	22	nfint 4920	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
24	21, 23	nffv 6918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})
25	24	nfel1 2942	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴
26		fveq2 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴} → (𝐹‘𝑦) = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}))
27	26	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴} → ((𝐹‘𝑦) ∈ 𝐴 ↔ (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴))
28	25, 27	onminsb 7832	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴 → (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴)
29	20, 28	eqeltrid 2868	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴 → 𝐷 ∈ 𝐴)
30	19, 29	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → 𝐷 ∈ 𝐴)
31	30	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → 𝐷 ∈ 𝐴)
32	4, 31	eqeltrd 2864	. . . . . . . . 9 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
33	32	3expia 1139	. . . . . . . 8 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴))
34		iftrue 4491	. . . . . . . . 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 2864	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
37	33, 36	pm2.61d2 183	. . . . . . 7 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴)
38		eleq1 2852	. . . . . . 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⟶1-1ontoV ∧ 𝐴 ≠ ∅) → (𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
40	39	rexlimdvw 3172	. . . . 5 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
41	40	abssdv 3995	. . . 4 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)} ⊆ 𝐴)
42	2, 41	eqsstrid 3951	. . 3 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → ran 𝐻 ⊆ 𝐴)
43		fveqeq2 6917	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑧) → ((𝐹‘𝑥) = 𝑧 ↔ (𝐹‘(◡𝐹‘𝑧)) = 𝑧))
44		f1ocnvdm 7316	. . . . . . . . . 10 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝑧 ∈ V) → (◡𝐹‘𝑧) ∈ On)
45	44	elvd 3460	. . . . . . . . 9 ⊢ (𝐹:On⟶1-1ontoV → (◡𝐹‘𝑧) ∈ On)
46		f1ocnvfv2 7308	. . . . . . . . . 10 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝑧 ∈ V) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
47	46	elvd 3460	. . . . . . . . 9 ⊢ (𝐹:On⟶1-1ontoV → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
48	43, 45, 47	rspcedvdw 3583	. . . . . . . 8 ⊢ (𝐹:On⟶1-1ontoV → ∃𝑥 ∈ On (𝐹‘𝑥) = 𝑧)
49		eleq1 2852	. . . . . . . . . . . . 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 4493	. . . . . . . . . . 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 2799	. . . . . . . . . 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 3181	. . . . . . . 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⟶1-1ontoV → (𝑧 ∈ 𝐴 → ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
57	56	ralrimiv 3157	. . . . . 6 ⊢ (𝐹:On⟶1-1ontoV → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
58		ssabral 3992	. . . . . 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⟶1-1ontoV → 𝐴 ⊆ {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)})
60	59, 2	sseqtrrdi 3954	. . . 4 ⊢ (𝐹:On⟶1-1ontoV → 𝐴 ⊆ ran 𝐻)
61	60	adantr 486	. . 3 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ ran 𝐻)
62	42, 61	eqssd 3930	. 2 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → ran 𝐻 = 𝐴)
63		fvex 6921	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
64	20	fvexi 6922	. . . . 5 ⊢ 𝐷 ∈ V
65	63, 64	ifex 4536	. . . 4 ⊢ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ V
66	65, 1	fnmpti 6704	. . 3 ⊢ 𝐻 Fn On
67		df-fo 6564	. . 3 ⊢ (𝐻:On⟶onto𝐴 ↔ (𝐻 Fn On ∧ ran 𝐻 = 𝐴))
68	66, 67	mpbiran 722	. 2 ⊢ (𝐻:On⟶onto𝐴 ↔ ran 𝐻 = 𝐴)
69	62, 68	sylibr 237	1 ⊢ ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → 𝐻:On⟶onto𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ifcif 4485  ∩ cint 4910   ↦ cmpt 5189  ^◡ccnv 5654  ran crn 5656  Oncon0 6380   Fn wfn 6553  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557  ‘cfv 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566

This theorem is used by: (None)