Theorem rankrelp 45902

Description: The rank function preserves ∈. (Contributed by Eric Schmidt, 11-Oct-2025.)

Assertion

Ref	Expression
rankrelp	⊢ rank RelPres E , E (∪ (𝑅1 “ On), On)

Proof of Theorem rankrelp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankf 9784	. 2 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		rankelb 9814	. . . . 5 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝑦 → (rank‘𝑥) ∈ (rank‘𝑦)))
3		epel 5554	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4		fvex 6890	. . . . . 6 ⊢ (rank‘𝑦) ∈ V
5	4	epeli 5553	. . . . 5 ⊢ ((rank‘𝑥) E (rank‘𝑦) ↔ (rank‘𝑥) ∈ (rank‘𝑦))
6	2, 3, 5	3imtr4g 299	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → (𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦)))
7	6	rgen 3079	. . 3 ⊢ ∀𝑦 ∈ ∪ (𝑅1 “ On)(𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦))
8	7	rgenw 3081	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)∀𝑦 ∈ ∪ (𝑅1 “ On)(𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦))
9		df-relp 45885	. 2 ⊢ (rank RelPres E , E (∪ (𝑅1 “ On), On) ↔ (rank:∪ (𝑅1 “ On)⟶On ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)∀𝑦 ∈ ∪ (𝑅1 “ On)(𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦))))
10	1, 8, 9	mpbir2an 724	1 ⊢ rank RelPres E , E (∪ (𝑅1 “ On), On)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3077  ∪ cuni 4867   class class class wbr 5103   E cep 5550   “ cima 5654  Oncon0 6355  ⟶wf 6527  ‘cfv 6531  𝑅₁cr1 9750  rankcrnk 9751   RelPres wrelp 45884

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-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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-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 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-r1 9752  df-rank 9753  df-relp 45885

This theorem is used by:  wffr  45903