Theorem vonf1osev 35604

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 35601	. 2 ⊢ (𝐹:V–1-1-onto→On → 𝑅 We V)
3		vex 3459	. . . . . . 7 ⊢ 𝑤 ∈ V
4		vex 3459	. . . . . . 7 ⊢ 𝑧 ∈ V
5		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
6	5	eleq1d 2848	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑦)))
7		fveq2 6881	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
8	7	eleq2d 2849	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ∈ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧)))
9	3, 4, 6, 8, 1	brab 5528	. . . . . 6 ⊢ (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧))
10		fvex 6894	. . . . . . 7 ⊢ (𝐹‘𝑧) ∈ V
11	10	epeli 5563	. . . . . 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 3084	. . . 4 ⊢ ∀𝑤 ∈ V ∀𝑧 ∈ V (𝑤𝑅𝑧 ↔ (𝐹‘𝑤) E (𝐹‘𝑧))
14		df-isom 6545	. . . 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 722	. . 3 ⊢ (𝐹 Isom 𝑅, E (V, On) ↔ 𝐹:V–1-1-onto→On)
16		epse 5643	. . . 4 ⊢ E Se On
17		isose 7341	. . . 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 520	1 ⊢ (𝐹:V–1-1-onto→On → (𝑅 We V ∧ 𝑅 Se V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   class class class wbr 5109  {copab 5173   E cep 5560   Se wse 5612   We wwe 5613  Oncon0 6360  –1-1-onto→wf1o 6535  ‘cfv 6536   Isom wiso 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545

This theorem is used by: (None)