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Theorem rankidb 9797
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 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄)))

Proof of Theorem rankidb
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 rankwflemb 9790 . . 3 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) ↔ βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜suc π‘₯))
2 nfcv 2903 . . . . . 6 β„²π‘₯𝑅1
3 nfrab1 3451 . . . . . . . 8 β„²π‘₯{π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}
43nfint 4960 . . . . . . 7 β„²π‘₯∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}
54nfsuc 6436 . . . . . 6 β„²π‘₯ suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}
62, 5nffv 6901 . . . . 5 β„²π‘₯(𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
76nfel2 2921 . . . 4 β„²π‘₯ 𝐴 ∈ (𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
8 suceq 6430 . . . . . 6 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)} β†’ suc π‘₯ = suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
98fveq2d 6895 . . . . 5 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)} β†’ (𝑅1β€˜suc π‘₯) = (𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
109eleq2d 2819 . . . 4 (π‘₯ = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)} β†’ (𝐴 ∈ (𝑅1β€˜suc π‘₯) ↔ 𝐴 ∈ (𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})))
117, 10onminsb 7784 . . 3 (βˆƒπ‘₯ ∈ On 𝐴 ∈ (𝑅1β€˜suc π‘₯) β†’ 𝐴 ∈ (𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
121, 11sylbi 216 . 2 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐴 ∈ (𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
13 rankvalb 9794 . . . 4 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
14 suceq 6430 . . . 4 ((rankβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)} β†’ suc (rankβ€˜π΄) = suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
1513, 14syl 17 . . 3 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ suc (rankβ€˜π΄) = suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)})
1615fveq2d 6895 . 2 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ (𝑅1β€˜suc (rankβ€˜π΄)) = (𝑅1β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 ∈ (𝑅1β€˜suc π‘₯)}))
1712, 16eleqtrrd 2836 1 (𝐴 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝐴 ∈ (𝑅1β€˜suc (rankβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  βˆͺ cuni 4908  βˆ© cint 4950   β€œ cima 5679  Oncon0 6364  suc csuc 6366  β€˜cfv 6543  π‘…1cr1 9759  rankcrnk 9760
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762
This theorem is referenced by:  rankdmr1  9798  rankr1ag  9799  sswf  9805  uniwf  9816  rankonidlem  9825  rankid  9830  dfac12lem2  10141  aomclem4  42101
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