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Theorem rankmapu 8779
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 7935 . . 3 (𝐴𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2 rankxpl.2 . . . . . 6 𝐵 ∈ V
3 rankxpl.1 . . . . . 6 𝐴 ∈ V
42, 3xpex 7004 . . . . 5 (𝐵 × 𝐴) ∈ V
54pwex 4878 . . . 4 𝒫 (𝐵 × 𝐴) ∈ V
65rankss 8750 . . 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 8744 . . 3 (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
92, 3rankxpu 8777 . . . . 5 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵𝐴))
10 uncom 3790 . . . . . . . 8 (𝐵𝐴) = (𝐴𝐵)
1110fveq2i 6232 . . . . . . 7 (rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵))
12 suceq 5828 . . . . . . 7 ((rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵)) → suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵)))
1311, 12ax-mp 5 . . . . . 6 suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵))
14 suceq 5828 . . . . . 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 3670 . . . 4 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵))
17 rankon 8696 . . . . . 6 (rank‘(𝐵 × 𝐴)) ∈ On
1817onordi 5870 . . . . 5 Ord (rank‘(𝐵 × 𝐴))
19 rankon 8696 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
2019onsuci 7080 . . . . . . 7 suc (rank‘(𝐴𝐵)) ∈ On
2120onsuci 7080 . . . . . 6 suc suc (rank‘(𝐴𝐵)) ∈ On
2221onordi 5870 . . . . 5 Ord suc suc (rank‘(𝐴𝐵))
23 ordsucsssuc 7065 . . . . 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 708 . . . 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 3668 . 2 (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
277, 26sstri 3645 1 (rank‘(𝐴𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607  𝒫 cpw 4191   × cxp 5141  Ord word 5760  suc csuc 5763  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-map 7901  df-pm 7902  df-r1 8665  df-rank 8666
This theorem is referenced by: (None)
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