Theorem vonf1oonfo 35422

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 5931	. . . 4 ⊢ ran 𝐻 = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)}
3		iffalse 4488	. . . . . . . . . . 11 ⊢ (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
4	3	3ad2ant3 1147	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅ ∧ ¬ (𝐹‘𝑥) ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = 𝐷)
5		n0 4305	. . . . . . . . . . . . 13 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐴)
6		19.42v 1972	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) ↔ (𝐹:On–1-1-onto→V ∧ ∃𝑤 𝑤 ∈ 𝐴))
7		f1ofo 6810	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On–1-1-onto→V → 𝐹:On–onto→V)
8		foelcdmi 6924	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:On–onto→V ∧ 𝑤 ∈ V) → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
9	8	elvd 3459	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:On–onto→V → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
10	7, 9	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:On–1-1-onto→V → ∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤)
11		r19.41v 3191	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐴)
14	13	reximi 3099	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑦 ∈ On ((𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
15	11, 14	sylbir 237	. . . . . . . . . . . . . . . 16 ⊢ ((∃𝑦 ∈ On (𝐹‘𝑦) = 𝑤 ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
16	10, 15	sylan 589	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
17	16	exlimiv 1949	. . . . . . . . . . . . . 14 ⊢ (∃𝑤(𝐹:On–1-1-onto→V ∧ 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
18	6, 17	sylbir 237	. . . . . . . . . . . . 13 ⊢ ((𝐹:On–1-1-onto→V ∧ ∃𝑤 𝑤 ∈ 𝐴) → ∃𝑦 ∈ On (𝐹‘𝑦) ∈ 𝐴)
19	5, 18	sylan2b 603	. . . . . . . . . . . 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 3433	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
23	22	nfint 4914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}
24	21, 23	nffv 6873	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})
25	24	nfel1 2939	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ∈ 𝐴
26		fveq2 6863	. . . . . . . . . . . . . . 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 7773	. . . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐷 ∈ 𝐴)
31	30	3adant3 1144	. . . . . . . . . 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 1133	. . . . . . . 8 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (¬ (𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ 𝐴))
34		iftrue 4485	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ 𝐴 → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = (𝐹‘𝑥))
35		id 22	. . . . . . . . 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 182	. . . . . . 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 249	. . . . . 6 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
40	39	rexlimdvw 3167	. . . . 5 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) → 𝑧 ∈ 𝐴))
41	40	abssdv 4020	. . . 4 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)} ⊆ 𝐴)
42	2, 41	eqsstrid 3974	. . 3 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ran 𝐻 ⊆ 𝐴)
43		fveqeq2 6872	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑧) → ((𝐹‘𝑥) = 𝑧 ↔ (𝐹‘(◡𝐹‘𝑧)) = 𝑧))
44		f1ocnvdm 7265	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑧 ∈ V) → (◡𝐹‘𝑧) ∈ On)
45	44	elvd 3459	. . . . . . . . 9 ⊢ (𝐹:On–1-1-onto→V → (◡𝐹‘𝑧) ∈ On)
46		f1ocnvfv2 7257	. . . . . . . . . 10 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝑧 ∈ V) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
47	46	elvd 3459	. . . . . . . . 9 ⊢ (𝐹:On–1-1-onto→V → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
48	43, 45, 47	rspcedvdw 3584	. . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐴)
51	50	iftrued 4487	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑧 ∧ 𝑧 ∈ 𝐴) → if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) = (𝐹‘𝑥))
52		simpl 486	. . . . . . . . . . 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 417	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑧 → 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
55	54	reximdv 3176	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (∃𝑥 ∈ On (𝐹‘𝑥) = 𝑧 → ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
56	48, 55	syl5com 31	. . . . . . 7 ⊢ (𝐹:On–1-1-onto→V → (𝑧 ∈ 𝐴 → ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)))
57	56	ralrimiv 3152	. . . . . 6 ⊢ (𝐹:On–1-1-onto→V → ∀𝑧 ∈ 𝐴 ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))
58		ssabral 4017	. . . . . 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 236	. . . . 5 ⊢ (𝐹:On–1-1-onto→V → 𝐴 ⊆ {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)})
60	59, 2	sseqtrrdi 3977	. . . 4 ⊢ (𝐹:On–1-1-onto→V → 𝐴 ⊆ ran 𝐻)
61	60	adantr 484	. . 3 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐴 ⊆ ran 𝐻)
62	42, 61	eqssd 3953	. 2 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → ran 𝐻 = 𝐴)
63		fvex 6876	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
64	20	fvexi 6877	. . . . 5 ⊢ 𝐷 ∈ V
65	63, 64	ifex 4530	. . . 4 ⊢ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷) ∈ V
66	65, 1	fnmpti 6660	. . 3 ⊢ 𝐻 Fn On
67		df-fo 6523	. . 3 ⊢ (𝐻:On–onto→𝐴 ↔ (𝐻 Fn On ∧ ran 𝐻 = 𝐴))
68	66, 67	mpbiran 719	. 2 ⊢ (𝐻:On–onto→𝐴 ↔ ran 𝐻 = 𝐴)
69	62, 68	sylibr 236	1 ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐻:On–onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  {crab 3413  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  ifcif 4479  ∩ cint 4904   ↦ cmpt 5180  ^◡ccnv 5644  ran crn 5646  Oncon0 6342   Fn wfn 6512  –onto→wfo 6515  –1-1-onto→wf1o 6516  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525

This theorem is referenced by: (None)