Theorem werankwe 35797

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 6890	. . . . . 6 ⊢ (𝑣 = 𝑥 → (rank‘𝑣) = (rank‘𝑥))
3	2	eqeq2d 2771	. . . . 5 ⊢ (𝑣 = 𝑥 → ((rank‘𝑧) = (rank‘𝑣) ↔ (rank‘𝑧) = (rank‘𝑥)))
4	3	rabbidv 3419	. . . 4 ⊢ (𝑣 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} = {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)})
5	1, 4	weeq12d 5640	. . 3 ⊢ (𝑣 = 𝑥 → (𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ↔ 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)}))
6	5	cbvralvw 3240	. 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 6903	. . . . . . 7 ⊢ (rank‘𝑦) ∈ V
10	9	epeli 5553	. . . . . 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 5171	. . . 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 2786	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
14		fveqeq2 6899	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((rank‘𝑧) = (rank‘𝑥) ↔ (rank‘𝑦) = (rank‘𝑥)))
15	14	cbvrabv 3422	. . . . . 6 ⊢ {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}
16	4, 15	eqtrdi 2811	. . . . 5 ⊢ (𝑣 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
17	1, 16	weeq12d 5640	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ↔ 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}))
18	17	rspccva 3575	. . 3 ⊢ ((∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ∧ 𝑥 ∈ 𝐴) → 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
19		rankfo 35793	. . . . . 6 ⊢ rank:V–onto→On
20		fof 6801	. . . . . 6 ⊢ (rank:V–onto→On → rank:V⟶On)
21	19, 20	ax-mp 5	. . . . 5 ⊢ rank:V⟶On
22		ssv 3933	. . . . 5 ⊢ 𝐴 ⊆ V
23		fssres 6753	. . . . 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 7793	. . . 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 8156	. 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 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3896   class class class wbr 5102  {copab 5166   E cep 5550   We wwe 5603   ↾ cres 5653  Oncon0 6365  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  rankcrnk 9774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-reg 9593  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-r1 9775  df-rank 9776

This theorem is used by:  onprcf1acwevdlem2  35997