Theorem werankwe 35862

Description: Construct a well-order given a relation 𝑅(𝑥) that well-orders elements of the same rank. (Contributed by BTernaryTau, 16-Sep-2026.)

Hypotheses

Ref	Expression
werankwe.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
werankwe.2	⊢ (𝑤 = (rank‘𝑥) → 𝑇 = 𝑅)
werankwe.3	⊢ (𝑣 = 𝑥 → 𝑈 = 𝑅)

Assertion

Ref	Expression
werankwe	⊢ (∀𝑥 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} → 𝑆 We 𝐴)

Distinct variable groups:   𝑥,𝑇,𝑦   𝑣,𝑅,𝑤,𝑦   𝑤,𝑈,𝑥,𝑦   𝑣,𝐴,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑅(𝑥, 𝑧)   𝑆(𝑥, 𝑦, 𝑧, 𝑤, 𝑣)   𝑇(𝑧, 𝑤, 𝑣)   𝑈(𝑧, 𝑣)

Proof of Theorem werankwe

Step	Hyp	Ref	Expression
1		werankwe.3	. . . 4 ⊢ (𝑣 = 𝑥 → 𝑈 = 𝑅)
2		fveq2 6916	. . . . . 6 ⊢ (𝑣 = 𝑥 → (rank‘𝑣) = (rank‘𝑥))
3	2	eqeq2d 2774	. . . . 5 ⊢ (𝑣 = 𝑥 → ((rank‘𝑧) = (rank‘𝑣) ↔ (rank‘𝑧) = (rank‘𝑥)))
4	3	rabbidv 3423	. . . 4 ⊢ (𝑣 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} = {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)})
5	1, 4	weeq12d 5647	. . 3 ⊢ (𝑣 = 𝑥 → (𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ↔ 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)}))
6	5	cbvralvw 3244	. 2 ⊢ (∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ↔ ∀𝑥 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)})
7		werankwe.2	. . 3 ⊢ (𝑤 = (rank‘𝑥) → 𝑇 = 𝑅)
8		werankwe.1	. . . 4 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
9		fvex 6929	. . . . . . 7 ⊢ (rank‘𝑦) ∈ V
10	9	epeli 5560	. . . . . 6 ⊢ ((rank‘𝑥) E (rank‘𝑦) ↔ (rank‘𝑥) ∈ (rank‘𝑦))
11	10	orbi1i 927	. . . . 5 ⊢ (((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦)))
12	11	opabbii 5178	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
13	8, 12	eqtr4i 2789	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
14		fveqeq2 6925	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((rank‘𝑧) = (rank‘𝑥) ↔ (rank‘𝑦) = (rank‘𝑥)))
15	14	cbvrabv 3426	. . . . . 6 ⊢ {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}
16	4, 15	eqtrdi 2814	. . . . 5 ⊢ (𝑣 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
17	1, 16	weeq12d 5647	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ↔ 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}))
18	17	rspccva 3579	. . 3 ⊢ ((∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ∧ 𝑥 ∈ 𝐴) → 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
19		rankfo 35858	. . . . . 6 ⊢ rank:V⟶ontoOn
20		fof 6826	. . . . . 6 ⊢ (rank:V⟶ontoOn → rank:V⟶On)
21	19, 20	ax-mp 5	. . . . 5 ⊢ rank:V⟶On
22		ssv 3937	. . . . 5 ⊢ 𝐴 ⊆ V
23		fssres 6778	. . . . 5 ⊢ ((rank:V⟶On ∧ 𝐴 ⊆ V) → (rank ↾ 𝐴):𝐴⟶On)
24	21, 22, 23	mp2an 705	. . . 4 ⊢ (rank ↾ 𝐴):𝐴⟶On
25	24	a1i 11	. . 3 ⊢ (∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} → (rank ↾ 𝐴):𝐴⟶On)
26		epweon 7821	. . . 4 ⊢ E We On
27	26	a1i 11	. . 3 ⊢ (∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} → E We On)
28	7, 13, 18, 25, 27	fnwe2 8186	. 2 ⊢ (∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} → 𝑆 We 𝐴)
29	6, 28	sylbir 238	1 ⊢ (∀𝑥 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} → 𝑆 We 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   class class class wbr 5108  {copab 5173   E cep 5557   We wwe 5610   ↾ cres 5660  Oncon0 6388  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  rankcrnk 9804

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-reg 9623  ax-inf2 9679

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-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  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-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-r1 9805  df-rank 9806

This theorem is used by:  onprcf1acwevdlem2  36087