Theorem rankidb 9023

Description: Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

Assertion

Ref	Expression
rankidb	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

Proof of Theorem rankidb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rankwflemb 9016	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2		nfcv 2933	. . . . . 6 ⊢ Ⅎ𝑥𝑅1
3		nfrab1 3325	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
4	3	nfint 4759	. . . . . . 7 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
5	4	nfsuc 6100	. . . . . 6 ⊢ Ⅎ𝑥 suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
6	2, 5	nffv 6509	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
7	6	nfel2 2949	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8		suceq 6094	. . . . . 6 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
9	8	fveq2d 6503	. . . . 5 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
10	9	eleq2d 2852	. . . 4 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})))
11	7, 10	onminsb 7330	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
12	1, 11	sylbi 209	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13		rankvalb 9020	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14		suceq 6094	. . . 4 ⊢ ((rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc (rank‘𝐴) = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
15	13, 14	syl 17	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
16	15	fveq2d 6503	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
17	12, 16	eleqtrrd 2870	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2050  ∃wrex 3090  {crab 3093  ∪ cuni 4712  ∩ cint 4749   “ cima 5410  Oncon0 6029  suc csuc 6031  ‘cfv 6188  𝑅₁cr1 8985  rankcrnk 8986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-om 7397  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-r1 8987  df-rank 8988

This theorem is referenced by:  rankdmr1  9024  rankr1ag  9025  sswf  9031  uniwf  9042  rankonidlem  9051  rankid  9056  dfac12lem2  9364  aomclem4  39050