Theorem vonf1osev 35419

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 35416	. 2 ⊢ (𝐹:V–1-1-onto→On → 𝑅 We V)
3		vex 3457	. . . . . . 7 ⊢ 𝑤 ∈ V
4		vex 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
5		fveq2 6863	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
6	5	eleq1d 2846	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑦)))
7		fveq2 6863	. . . . . . . 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 5512	. . . . . 6 ⊢ (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧))
10		fvex 6876	. . . . . . 7 ⊢ (𝐹‘𝑧) ∈ V
11	10	epeli 5547	. . . . . 6 ⊢ ((𝐹‘𝑤) E (𝐹‘𝑧) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧))
12	9, 11	bitr4i 280	. . . . 5 ⊢ (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))
13	12	rgen2w 3080	. . . 4 ⊢ ∀𝑤 ∈ V ∀𝑧 ∈ V (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))
14		df-isom 6526	. . . 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 720	. . 3 ⊢ (𝐹 Isom 𝑅, E (V, On) ↔ 𝐹:V–1-1-onto→On)
16		epse 5627	. . . 4 ⊢ E Se On
17		isose 7323	. . . 4 ⊢ (𝐹 Isom 𝑅, E (V, On) → (𝑅 Se V ↔ E Se On))
18	16, 17	mpbiri 260	. . 3 ⊢ (𝐹 Isom 𝑅, E (V, On) → 𝑅 Se V)
19	15, 18	sylbir 237	. 2 ⊢ (𝐹:V–1-1-onto→On → 𝑅 Se V)
20	2, 19	jca 519	1 ⊢ (𝐹:V–1-1-onto→On → (𝑅 We V ∧ 𝑅 Se V))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   class class class wbr 5099  {copab 5161   E cep 5544   Se wse 5596   We wwe 5597  Oncon0 6342  –1-1-onto→wf1o 6516  ‘cfv 6517   Isom wiso 6518

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-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

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-tp 4586  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-se 5599  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  df-isom 6526

This theorem is referenced by: (None)