Theorem werankwe2 35822

Description: Construct a well-order given a relation that well-orders elements of the same rank. See werankwe 35821 for a version where 𝑅 can depend on 𝑤. (Contributed by BTernaryTau, 18-Sep-2026.)

Hypothesis

Ref	Expression
werankwe2.1	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}

Assertion

Ref	Expression
werankwe2	⊢ (∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} → 𝑆 We 𝐴)

Distinct variable groups:   𝑤,𝑅,𝑥,𝑦   𝑤,𝐴,𝑥,𝑦,𝑧

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

Proof of Theorem werankwe2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2764	. 2 ⊢ (𝑣 = (rank‘𝑥) → 𝑅 = 𝑅)
2		werankwe2.1	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
3		fvex 6921	. . . . . 6 ⊢ (rank‘𝑦) ∈ V
4	3	epeli 5557	. . . . 5 ⊢ ((rank‘𝑥) E (rank‘𝑦) ↔ (rank‘𝑥) ∈ (rank‘𝑦))
5	4	orbi1i 927	. . . 4 ⊢ (((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦)) ↔ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦)))
6	5	opabbii 5175	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
7	2, 6	eqtr4i 2789	. 2 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
8		eqidd 2764	. . . 4 ⊢ (𝑤 = 𝑥 → 𝑅 = 𝑅)
9		fveq2 6908	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
10	9	eqeq2d 2774	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((rank‘𝑧) = (rank‘𝑤) ↔ (rank‘𝑧) = (rank‘𝑥)))
11	10	rabbidv 3423	. . . . 5 ⊢ (𝑤 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} = {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)})
12		fveqeq2 6917	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((rank‘𝑧) = (rank‘𝑥) ↔ (rank‘𝑦) = (rank‘𝑥)))
13	12	cbvrabv 3426	. . . . 5 ⊢ {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}
14	11, 13	eqtrdi 2814	. . . 4 ⊢ (𝑤 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
15	8, 14	weeq12d 5644	. . 3 ⊢ (𝑤 = 𝑥 → (𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} ↔ 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}))
16	15	rspccva 3579	. 2 ⊢ ((∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} ∧ 𝑥 ∈ 𝐴) → 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
17		rankfo 35817	. . . . 5 ⊢ rank:V⟶ontoOn
18		fof 6818	. . . . 5 ⊢ (rank:V⟶ontoOn → rank:V⟶On)
19	17, 18	ax-mp 5	. . . 4 ⊢ rank:V⟶On
20		ssv 3937	. . . 4 ⊢ 𝐴 ⊆ V
21		fssres 6770	. . . 4 ⊢ ((rank:V⟶On ∧ 𝐴 ⊆ V) → (rank ↾ 𝐴):𝐴⟶On)
22	19, 20, 21	mp2an 705	. . 3 ⊢ (rank ↾ 𝐴):𝐴⟶On
23	22	a1i 11	. 2 ⊢ (∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} → (rank ↾ 𝐴):𝐴⟶On)
24		epweon 7813	. . 3 ⊢ E We On
25	24	a1i 11	. 2 ⊢ (∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} → E We On)
26	1, 7, 16, 23, 25	fnwe2 8176	1 ⊢ (∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} → 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 5106  {copab 5170   E cep 5554   We wwe 5607   ↾ cres 5657  Oncon0 6380  ⟶wf 6554  ⟶ontowfo 6556  ‘cfv 6558  rankcrnk 9794

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-pow 5331  ax-pr 5395  ax-un 7775  ax-reg 9613  ax-inf2 9669

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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  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-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  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-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-r1 9795  df-rank 9796

This theorem is used by:  wevsev  36041