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Theorem rankmapu 9915
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 8916 . . 3 (𝐴m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2 rankxpl.2 . . . . . 6 𝐵 ∈ V
3 rankxpl.1 . . . . . 6 𝐴 ∈ V
42, 3xpex 7771 . . . . 5 (𝐵 × 𝐴) ∈ V
54pwex 5385 . . . 4 𝒫 (𝐵 × 𝐴) ∈ V
65rankss 9886 . . 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 9880 . . 3 (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
92, 3rankxpu 9913 . . . . 5 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵𝐴))
10 uncom 4167 . . . . . . . 8 (𝐵𝐴) = (𝐴𝐵)
1110fveq2i 6909 . . . . . . 7 (rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵))
12 suceq 6451 . . . . . . 7 ((rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵)) → suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵)))
1311, 12ax-mp 5 . . . . . 6 suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵))
14 suceq 6451 . . . . . 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 4031 . . . 4 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵))
17 rankon 9832 . . . . . 6 (rank‘(𝐵 × 𝐴)) ∈ On
1817onordi 6496 . . . . 5 Ord (rank‘(𝐵 × 𝐴))
19 rankon 9832 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
2019onsuci 7858 . . . . . . 7 suc (rank‘(𝐴𝐵)) ∈ On
2120onsuci 7858 . . . . . 6 suc suc (rank‘(𝐴𝐵)) ∈ On
2221onordi 6496 . . . . 5 Ord suc suc (rank‘(𝐴𝐵))
23 ordsucsssuc 7842 . . . . 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 692 . . . 4 ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵)))
2516, 24mpbi 230 . . 3 suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
268, 25eqsstri 4029 . 2 (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
277, 26sstri 4004 1 (rank‘(𝐴m 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  wcel 2105  Vcvv 3477  cun 3960  wss 3962  𝒫 cpw 4604   × cxp 5686  Ord word 6384  suc csuc 6387  cfv 6562  (class class class)co 7430  m cmap 8864  rankcrnk 9800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-map 8866  df-pm 8867  df-r1 9801  df-rank 9802
This theorem is referenced by: (None)
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