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Theorem rankmapu 9095
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 8236 . . 3 (𝐴𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2 rankxpl.2 . . . . . 6 𝐵 ∈ V
3 rankxpl.1 . . . . . 6 𝐴 ∈ V
42, 3xpex 7287 . . . . 5 (𝐵 × 𝐴) ∈ V
54pwex 5128 . . . 4 𝒫 (𝐵 × 𝐴) ∈ V
65rankss 9066 . . 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 9060 . . 3 (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
92, 3rankxpu 9093 . . . . 5 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵𝐴))
10 uncom 4012 . . . . . . . 8 (𝐵𝐴) = (𝐴𝐵)
1110fveq2i 6496 . . . . . . 7 (rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵))
12 suceq 6088 . . . . . . 7 ((rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵)) → suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵)))
1311, 12ax-mp 5 . . . . . 6 suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵))
14 suceq 6088 . . . . . 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 3887 . . . 4 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵))
17 rankon 9012 . . . . . 6 (rank‘(𝐵 × 𝐴)) ∈ On
1817onordi 6127 . . . . 5 Ord (rank‘(𝐵 × 𝐴))
19 rankon 9012 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
2019onsuci 7363 . . . . . . 7 suc (rank‘(𝐴𝐵)) ∈ On
2120onsuci 7363 . . . . . 6 suc suc (rank‘(𝐴𝐵)) ∈ On
2221onordi 6127 . . . . 5 Ord suc suc (rank‘(𝐴𝐵))
23 ordsucsssuc 7348 . . . . 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 679 . . . 4 ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵)))
2516, 24mpbi 222 . . 3 suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
268, 25eqsstri 3885 . 2 (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
277, 26sstri 3861 1 (rank‘(𝐴𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1507  wcel 2050  Vcvv 3409  cun 3821  wss 3823  𝒫 cpw 4416   × cxp 5399  Ord word 6022  suc csuc 6025  cfv 6182  (class class class)co 6970  𝑚 cmap 8200  rankcrnk 8980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-reg 8845  ax-inf2 8892
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7495  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-map 8202  df-pm 8203  df-r1 8981  df-rank 8982
This theorem is referenced by: (None)
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