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Theorem mappwen 8973
 Description: Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
mappwen (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)

Proof of Theorem mappwen
StepHypRef Expression
1 simprr 811 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2 pw2eng 8107 . . . . . 6 (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
32ad2antrr 762 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
4 domentr 8056 . . . . 5 ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
51, 3, 4syl2anc 694 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
6 mapdom1 8166 . . . 4 (𝐴 ≼ (2𝑜𝑚 𝐵) → (𝐴𝑚 𝐵) ≼ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵))
75, 6syl 17 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≼ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵))
8 2on 7613 . . . . . . 7 2𝑜 ∈ On
98a1i 11 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 2𝑜 ∈ On)
10 simpll 805 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
11 mapxpen 8167 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
129, 10, 10, 11syl3anc 1366 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
138elexi 3244 . . . . . . 7 2𝑜 ∈ V
1413enref 8030 . . . . . 6 2𝑜 ≈ 2𝑜
15 infxpidm2 8878 . . . . . . 7 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵) → (𝐵 × 𝐵) ≈ 𝐵)
1615adantr 480 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐵 × 𝐵) ≈ 𝐵)
17 mapen 8165 . . . . . 6 ((2𝑜 ≈ 2𝑜 ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
1814, 16, 17sylancr 696 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
19 entr 8049 . . . . 5 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)) ∧ (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
2012, 18, 19syl2anc 694 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
213ensymd 8048 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵)
22 entr 8049 . . . 4 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
2320, 21, 22syl2anc 694 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
24 domentr 8056 . . 3 (((𝐴𝑚 𝐵) ≼ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
257, 23, 24syl2anc 694 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
26 mapdom1 8166 . . . 4 (2𝑜𝐴 → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
2726ad2antrl 764 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
28 endomtr 8055 . . 3 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
293, 27, 28syl2anc 694 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
30 sbth 8121 . 2 (((𝐴𝑚 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
3125, 29, 30syl2anc 694 1 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030  𝒫 cpw 4191   class class class wbr 4685   × cxp 5141  dom cdm 5143  Oncon0 5761  (class class class)co 6690  ωcom 7107  2𝑜c2o 7599   ↑𝑚 cmap 7899   ≈ cen 7994   ≼ cdom 7995  cardccrd 8799 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-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-rmo 2949  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-se 5103  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-isom 5935  df-riota 6651  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-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803 This theorem is referenced by:  alephexp1  9439  hauspwdom  21352
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