Theorem onf1we 35803

Description: If 𝐹 maps 𝐴 one-to-one into the ordinals, then 𝑅 well-orders 𝐴. (Contributed by BTernaryTau, 16-Sep-2026.)

Hypothesis

Ref	Expression
onf1we.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}

Assertion

Ref	Expression
onf1we	⊢ (𝐹:𝐴⟶1-1On → 𝑅 We 𝐴)

Distinct variable group:   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝑅(𝑥, 𝑦)

Proof of Theorem onf1we

Step	Hyp	Ref	Expression
1		f1f 6800	. . . 4 ⊢ (𝐹:𝐴⟶1-1On → 𝐹:𝐴⟶On)
2	1	frnd 6740	. . 3 ⊢ (𝐹:𝐴⟶1-1On → ran 𝐹 ⊆ On)
3		epweon 7813	. . 3 ⊢ E We On
4		wess 5641	. . 3 ⊢ (ran 𝐹 ⊆ On → ( E We On → E We ran 𝐹))
5	2, 3, 4	mpisyl 22	. 2 ⊢ (𝐹:𝐴⟶1-1On → E We ran 𝐹)
6		f1f1orn 6858	. . 3 ⊢ (𝐹:𝐴⟶1-1On → 𝐹:𝐴⟶1-1ontoran 𝐹)
7		onf1we.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}
8		fvex 6921	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
9	8	epeli 5557	. . . . . 6 ⊢ ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦))
10	9	opabbii 5175	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) E (𝐹‘𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}
11	7, 10	eqtr4i 2789	. . . 4 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) E (𝐹‘𝑦)}
12	11	f1owe 7384	. . 3 ⊢ (𝐹:𝐴⟶1-1ontoran 𝐹 → (𝑅 We 𝐴 ↔ E We ran 𝐹))
13	6, 12	syl 18	. 2 ⊢ (𝐹:𝐴⟶1-1On → (𝑅 We 𝐴 ↔ E We ran 𝐹))
14	5, 13	mpbird 260	1 ⊢ (𝐹:𝐴⟶1-1On → 𝑅 We 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106  {copab 5170   E cep 5554   We wwe 5607  ran crn 5656  Oncon0 6380  ⟶1-1wf1 6555  ⟶1-1ontowf1o 6557  ‘cfv 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567

This theorem is used by:  dfac15  35845