Theorem rankidb 9416

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 9409	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥𝑅1
3		nfrab1 3296	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
4	3	nfint 4869	. . . . . . 7 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
5	4	nfsuc 6284	. . . . . 6 ⊢ Ⅎ𝑥 suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
6	2, 5	nffv 6727	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
7	6	nfel2 2922	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8		suceq 6278	. . . . . 6 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
9	8	fveq2d 6721	. . . . 5 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
10	9	eleq2d 2823	. . . 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 7578	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
12	1, 11	sylbi 220	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13		rankvalb 9413	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14		suceq 6278	. . . 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 6721	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
17	12, 16	eleqtrrd 2841	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  ∃wrex 3062  {crab 3065  ∪ cuni 4819  ∩ cint 4859   “ cima 5554  Oncon0 6213  suc csuc 6215  ‘cfv 6380  𝑅₁cr1 9378  rankcrnk 9379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-r1 9380  df-rank 9381

This theorem is referenced by:  rankdmr1  9417  rankr1ag  9418  sswf  9424  uniwf  9435  rankonidlem  9444  rankid  9449  dfac12lem2  9758  aomclem4  40585