Theorem werankwe2 35785

Description: Construct a well-order given a relation that well-orders elements of the same rank. See werankwe 35784 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 2761	. 2 ⊢ (𝑣 = (rank‘𝑥) → 𝑅 = 𝑅)
2		werankwe2.1	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
3		fvex 6898	. . . . . 6 ⊢ (rank‘𝑦) ∈ V
4	3	epeli 5552	. . . . 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 5171	. . 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 2786	. 2 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) E (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}
8		eqidd 2761	. . . 4 ⊢ (𝑤 = 𝑥 → 𝑅 = 𝑅)
9		fveq2 6885	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
10	9	eqeq2d 2771	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((rank‘𝑧) = (rank‘𝑤) ↔ (rank‘𝑧) = (rank‘𝑥)))
11	10	rabbidv 3419	. . . . 5 ⊢ (𝑤 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} = {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)})
12		fveqeq2 6894	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((rank‘𝑧) = (rank‘𝑥) ↔ (rank‘𝑦) = (rank‘𝑥)))
13	12	cbvrabv 3422	. . . . 5 ⊢ {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}
14	11, 13	eqtrdi 2811	. . . 4 ⊢ (𝑤 = 𝑥 → {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} = {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
15	8, 14	weeq12d 5639	. . 3 ⊢ (𝑤 = 𝑥 → (𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} ↔ 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)}))
16	15	rspccva 3575	. 2 ⊢ ((∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} ∧ 𝑥 ∈ 𝐴) → 𝑅 We {𝑦 ∈ 𝐴 ∣ (rank‘𝑦) = (rank‘𝑥)})
17		rankfo 35780	. . . . 5 ⊢ rank:V–onto→On
18		fof 6796	. . . . 5 ⊢ (rank:V–onto→On → rank:V⟶On)
19	17, 18	ax-mp 5	. . . 4 ⊢ rank:V⟶On
20		ssv 3932	. . . 4 ⊢ 𝐴 ⊆ V
21		fssres 6748	. . . 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 7789	. . 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 8149	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 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102  {copab 5166   E cep 5549   We wwe 5602   ↾ cres 5652  Oncon0 6362  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  rankcrnk 9768

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 5326  ax-pr 5390  ax-un 7751  ax-reg 9587  ax-inf2 9643

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-r1 9769  df-rank 9770

This theorem is used by:  wevsev  35953