Theorem vonf1osev 35717

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 35714	. 2 ⊢ (𝐹:V–1-1-onto→On → 𝑅 We V)
3		vex 3457	. . . . . . 7 ⊢ 𝑤 ∈ V
4		vex 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
5		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
6	5	eleq1d 2847	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑦)))
7		fveq2 6882	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
8	7	eleq2d 2848	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧)))
9	3, 4, 6, 8, 1	brab 5526	. . . . . 6 ⊢ (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧))
10		fvex 6895	. . . . . . 7 ⊢ (𝐹‘𝑧) ∈ V
11	10	epeli 5561	. . . . . 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 3083	. . . 4 ⊢ ∀𝑤 ∈ V ∀𝑧 ∈ V (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))
14		df-isom 6546	. . . 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 5641	. . . 4 ⊢ E Se On
17		isose 7348	. . . 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 3078  Vcvv 3453   class class class wbr 5107  {copab 5171   E cep 5558   Se wse 5610   We wwe 5611  Oncon0 6361  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546

This theorem is used by: (None)