Theorem werankwe 35739

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 6885	. . . . . 6 ⊢ (𝑣 = 𝑥 → (rank‘𝑣) = (rank‘𝑥))
3	2	eqeq2d 2772	. . . . 5 ⊢ (𝑣 = 𝑥 → ((rank‘𝑧) = (rank‘𝑣) ↔ (rank‘𝑧) = (rank‘𝑥)))
4	3	rabbidv 3420	. . . 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 3241	. 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 6898	. . . . . . 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 5172	. . . 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 2787	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
14		fveqeq2 6894	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((rank‘𝑧) = (rank‘𝑥) ↔ (rank‘𝑦) = (rank‘𝑥)))
15	14	cbvrabv 3423	. . . . . 6 ⊢ {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}
16	4, 15	eqtrdi 2812	. . . . 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 3576	. . 3 ⊢ ((∀𝑣 ∈ 𝐴 𝑈 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑣)} ∧ 𝑥 ∈ 𝐴) → 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
19		rankfo 35735	. . . . . 6 ⊢ rank:V–onto→On
20		fof 6796	. . . . . 6 ⊢ (rank:V–onto→On → rank:V⟶On)
21	19, 20	ax-mp 5	. . . . 5 ⊢ rank:V⟶On
22		ssv 3955	. . . . 5 ⊢ 𝐴 ⊆ V
23		fssres 6748	. . . . 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 7789	. . . 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 8149	. 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 2145  ∀wral 3077  {crab 3413  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103  {copab 5167   E cep 5550   We wwe 5603   ↾ cres 5653  Oncon0 6362  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  rankcrnk 9767

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769

This theorem is used by:  onprcf1acwevdlem2  35896