Theorem rankrelp 45405

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 9709	. 2 ⊢ rank:∪ (𝑅1 “ On)⟶On
2		rankelb 9739	. . . . 5 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝑦 → (rank‘𝑥) ∈ (rank‘𝑦)))
3		epel 5527	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4		fvex 6847	. . . . . 6 ⊢ (rank‘𝑦) ∈ V
5	4	epeli 5526	. . . . 5 ⊢ ((rank‘𝑥) E (rank‘𝑦) ↔ (rank‘𝑥) ∈ (rank‘𝑦))
6	2, 3, 5	3imtr4g 296	. . . 4 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → (𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦)))
7	6	rgen 3054	. . 3 ⊢ ∀𝑦 ∈ ∪ (𝑅1 “ On)(𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦))
8	7	rgenw 3056	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)∀𝑦 ∈ ∪ (𝑅1 “ On)(𝑥 E 𝑦 → (rank‘𝑥) E (rank‘𝑦))
9		df-relp 45388	. 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 712	1 ⊢ rank RelPres E , E (∪ (𝑅1 “ On), On)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052  ∪ cuni 4851   class class class wbr 5086   E cep 5523   “ cima 5627  Oncon0 6317  ⟶wf 6488  ‘cfv 6492  𝑅₁cr1 9677  rankcrnk 9678   RelPres wrelp 45387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-r1 9679  df-rank 9680  df-relp 45388

This theorem is referenced by:  wffr  45406