MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankmapu Structured version   Visualization version   GIF version

Theorem rankmapu 8686
Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)
Hypotheses
Ref Expression
rankxpl.1 𝐴 ∈ V
rankxpl.2 𝐵 ∈ V
Assertion
Ref Expression
rankmapu (rank‘(𝐴𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))

Proof of Theorem rankmapu
StepHypRef Expression
1 mapsspw 7838 . . 3 (𝐴𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2 rankxpl.2 . . . . . 6 𝐵 ∈ V
3 rankxpl.1 . . . . . 6 𝐴 ∈ V
42, 3xpex 6916 . . . . 5 (𝐵 × 𝐴) ∈ V
54pwex 4813 . . . 4 𝒫 (𝐵 × 𝐴) ∈ V
65rankss 8657 . . 3 ((𝐴𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) → (rank‘(𝐴𝑚 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴)))
71, 6ax-mp 5 . 2 (rank‘(𝐴𝑚 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴))
84rankpw 8651 . . 3 (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
92, 3rankxpu 8684 . . . . 5 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵𝐴))
10 uncom 3740 . . . . . . . 8 (𝐵𝐴) = (𝐴𝐵)
1110fveq2i 6153 . . . . . . 7 (rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵))
12 suceq 5752 . . . . . . 7 ((rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵)) → suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵)))
1311, 12ax-mp 5 . . . . . 6 suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵))
14 suceq 5752 . . . . . 6 (suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵)) → suc suc (rank‘(𝐵𝐴)) = suc suc (rank‘(𝐴𝐵)))
1513, 14ax-mp 5 . . . . 5 suc suc (rank‘(𝐵𝐴)) = suc suc (rank‘(𝐴𝐵))
169, 15sseqtri 3621 . . . 4 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵))
17 rankon 8603 . . . . . 6 (rank‘(𝐵 × 𝐴)) ∈ On
1817onordi 5794 . . . . 5 Ord (rank‘(𝐵 × 𝐴))
19 rankon 8603 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
2019onsuci 6986 . . . . . . 7 suc (rank‘(𝐴𝐵)) ∈ On
2120onsuci 6986 . . . . . 6 suc suc (rank‘(𝐴𝐵)) ∈ On
2221onordi 5794 . . . . 5 Ord suc suc (rank‘(𝐴𝐵))
23 ordsucsssuc 6971 . . . . 5 ((Ord (rank‘(𝐵 × 𝐴)) ∧ Ord suc suc (rank‘(𝐴𝐵))) → ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))))
2418, 22, 23mp2an 707 . . . 4 ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵)))
2516, 24mpbi 220 . . 3 suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
268, 25eqsstri 3619 . 2 (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
277, 26sstri 3597 1 (rank‘(𝐴𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1992  Vcvv 3191  cun 3558  wss 3560  𝒫 cpw 4135   × cxp 5077  Ord word 5684  suc csuc 5687  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  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-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-reg 8442  ax-inf2 8483
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-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-map 7805  df-pm 7806  df-r1 8572  df-rank 8573
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator