Theorem vonf1osev 35864

Description: If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 is a set-like well-ordering of the universe. This is the ZFC version of (2 → 4) which is used in place of (3 → 4) in https://tinyurl.com/hamkins-gblac. This proof takes advantage of the fact that the well-order constructed in (2 → 3) is also set-like. (Contributed by BTernaryTau, 8-Jun-2026.)

Hypothesis

Ref	Expression
vonf1osev.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}

Assertion

Ref	Expression
vonf1osev	⊢ (𝐹:V–1-1-onto→On → (𝑅 We V ∧ 𝑅 Se V))

Distinct variable group:   𝑥,𝐹,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem vonf1osev

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

Step	Hyp	Ref	Expression
1		vonf1osev.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}
2	1	vonf1owev 35861	. 2 ⊢ (𝐹:V–1-1-onto→On → 𝑅 We V)
3		vex 3455	. . . . . . 7 ⊢ 𝑤 ∈ V
4		vex 3455	. . . . . . 7 ⊢ 𝑧 ∈ V
5		fveq2 6877	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
6	5	eleq1d 2846	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑦)))
7		fveq2 6877	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
8	7	eleq2d 2847	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧)))
9	3, 4, 6, 8, 1	brab 5518	. . . . . 6 ⊢ (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧))
10		fvex 6890	. . . . . . 7 ⊢ (𝐹‘𝑧) ∈ V
11	10	epeli 5553	. . . . . 6 ⊢ ((𝐹‘𝑤) E (𝐹‘𝑧) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧))
12	9, 11	bitr4i 281	. . . . 5 ⊢ (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))
13	12	rgen2w 3082	. . . 4 ⊢ ∀𝑤 ∈ V ∀𝑧 ∈ V (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))
14		df-isom 6540	. . . 4 ⊢ (𝐹 Isom 𝑅, E (V, On) ↔ (𝐹:V–1-1-onto→On ∧ ∀𝑤 ∈ V ∀𝑧 ∈ V (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))))
15	13, 14	mpbiran2 723	. . 3 ⊢ (𝐹 Isom 𝑅, E (V, On) ↔ 𝐹:V–1-1-onto→On)
16		epse 5633	. . . 4 ⊢ E Se On
17		isose 7343	. . . 4 ⊢ (𝐹 Isom 𝑅, E (V, On) → (𝑅 Se V ↔ E Se On))
18	16, 17	mpbiri 261	. . 3 ⊢ (𝐹 Isom 𝑅, E (V, On) → 𝑅 Se V)
19	15, 18	sylbir 238	. 2 ⊢ (𝐹:V–1-1-onto→On → 𝑅 Se V)
20	2, 19	jca 521	1 ⊢ (𝐹:V–1-1-onto→On → (𝑅 We V ∧ 𝑅 Se V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   class class class wbr 5103  {copab 5167   E cep 5550   Se wse 5602   We wwe 5603  Oncon0 6355  –1-1-onto→wf1o 6530  ‘cfv 6531   Isom wiso 6532

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-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

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-tp 4589  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-se 5605  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  df-isom 6540

This theorem is used by: (None)