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Theorem rankmapu 9293
 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 8427 . . 3 (𝐴m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2 rankxpl.2 . . . . . 6 𝐵 ∈ V
3 rankxpl.1 . . . . . 6 𝐴 ∈ V
42, 3xpex 7458 . . . . 5 (𝐵 × 𝐴) ∈ V
54pwex 5246 . . . 4 𝒫 (𝐵 × 𝐴) ∈ V
65rankss 9264 . . 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 9258 . . 3 (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
92, 3rankxpu 9291 . . . . 5 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵𝐴))
10 uncom 4080 . . . . . . . 8 (𝐵𝐴) = (𝐴𝐵)
1110fveq2i 6648 . . . . . . 7 (rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵))
12 suceq 6224 . . . . . . 7 ((rank‘(𝐵𝐴)) = (rank‘(𝐴𝐵)) → suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵)))
1311, 12ax-mp 5 . . . . . 6 suc (rank‘(𝐵𝐴)) = suc (rank‘(𝐴𝐵))
14 suceq 6224 . . . . . 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 3951 . . . 4 (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵))
17 rankon 9210 . . . . . 6 (rank‘(𝐵 × 𝐴)) ∈ On
1817onordi 6263 . . . . 5 Ord (rank‘(𝐵 × 𝐴))
19 rankon 9210 . . . . . . . 8 (rank‘(𝐴𝐵)) ∈ On
2019onsuci 7535 . . . . . . 7 suc (rank‘(𝐴𝐵)) ∈ On
2120onsuci 7535 . . . . . 6 suc suc (rank‘(𝐴𝐵)) ∈ On
2221onordi 6263 . . . . 5 Ord suc suc (rank‘(𝐴𝐵))
23 ordsucsssuc 7520 . . . . 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 691 . . . 4 ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵)))
2516, 24mpbi 233 . . 3 suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
268, 25eqsstri 3949 . 2 (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴𝐵))
277, 26sstri 3924 1 (rank‘(𝐴m 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  𝒫 cpw 4497   × cxp 5517  Ord word 6158  suc csuc 6161  ‘cfv 6324  (class class class)co 7135   ↑m cmap 8391  rankcrnk 9178 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-reg 9042  ax-inf2 9090 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-map 8393  df-pm 8394  df-r1 9179  df-rank 9180 This theorem is referenced by: (None)
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