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Theorem rankidb 8608
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.
StepHypRef Expression
1 rankwflemb 8601 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2 nfcv 2767 . . . . . 6 𝑥𝑅1
3 nfrab1 3116 . . . . . . . 8 𝑥{𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
43nfint 4456 . . . . . . 7 𝑥 {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
54nfsuc 5758 . . . . . 6 𝑥 suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}
62, 5nffv 6157 . . . . 5 𝑥(𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
76nfel2 2783 . . . 4 𝑥 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
8 suceq 5752 . . . . . 6 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc 𝑥 = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
98fveq2d 6154 . . . . 5 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝑅1‘suc 𝑥) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
109eleq2d 2689 . . . 4 (𝑥 = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})))
117, 10onminsb 6947 . . 3 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
121, 11sylbi 207 . 2 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
13 rankvalb 8605 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
14 suceq 5752 . . . 4 ((rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → suc (rank‘𝐴) = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
1513, 14syl 17 . . 3 (𝐴 (𝑅1 “ On) → suc (rank‘𝐴) = suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
1615fveq2d 6154 . 2 (𝐴 (𝑅1 “ On) → (𝑅1‘suc (rank‘𝐴)) = (𝑅1‘suc {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}))
1712, 16eleqtrrd 2707 1 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  wrex 2913  {crab 2916   cuni 4407   cint 4445  cima 5082  Oncon0 5685  suc csuc 5687  cfv 5850  𝑅1cr1 8570  rankcrnk 8571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-r1 8572  df-rank 8573
This theorem is referenced by:  rankdmr1  8609  rankr1ag  8610  sswf  8616  uniwf  8627  rankonidlem  8636  rankid  8641  dfac12lem2  8911  aomclem4  37074
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