Theorem rankidb 9832

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 9824	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥𝑅1
3		nfrab1 3435	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
4	3	nfint 4920	. . . . . . 7 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
5	4	nfsuc 6452	. . . . . 6 ⊢ Ⅎ𝑥 suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
6	2, 5	nffv 6915	. . . . 5 ⊢ Ⅎ𝑥(𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
7	6	nfel2 2944	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8		suceq 6446	. . . . . 6 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
9	8	fveq2d 6909	. . . . 5 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
10	9	eleq2d 2850	. . . 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 7829	. . 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 9829	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14		suceq 6446	. . . 4 ⊢ ((rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc (rank‘𝐴) = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
15	13, 14	syl 18	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → suc (rank‘𝐴) = suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
16	15	fveq2d 6909	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
17	12, 16	eleqtrrd 2867	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  ∪ cuni 4870  ∩ cint 4910   “ cima 5658  Oncon0 6377  suc csuc 6379  ‘cfv 6555  𝑅₁cr1 9790  rankcrnk 9791

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792  df-rank 9793

This theorem is used by:  rankdmr1  9833  rankr1ag  9834  sswf  9840  uniwf  9852  rankonidlem  9862  rankid  9869  rankval4b  9904  r1filimi  9927  dfac12lem2  10247  aomclem4  44250