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Theorem mappwen 9135
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 756 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2 pw2eng 8222 . . . . . 6 (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
32ad2antrr 705 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
4 domentr 8168 . . . . 5 ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
51, 3, 4syl2anc 573 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
6 mapdom1 8281 . . . 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 7722 . . . . . . 7 2𝑜 ∈ On
98a1i 11 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 2𝑜 ∈ On)
10 simpll 750 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
11 mapxpen 8282 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
129, 10, 10, 11syl3anc 1476 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
138elexi 3365 . . . . . . 7 2𝑜 ∈ V
1413enref 8142 . . . . . 6 2𝑜 ≈ 2𝑜
15 infxpidm2 9040 . . . . . . 7 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵) → (𝐵 × 𝐵) ≈ 𝐵)
1615adantr 466 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐵 × 𝐵) ≈ 𝐵)
17 mapen 8280 . . . . . 6 ((2𝑜 ≈ 2𝑜 ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
1814, 16, 17sylancr 575 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
19 entr 8161 . . . . 5 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)) ∧ (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
2012, 18, 19syl2anc 573 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
213ensymd 8160 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵)
22 entr 8161 . . . 4 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
2320, 21, 22syl2anc 573 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
24 domentr 8168 . . 3 (((𝐴𝑚 𝐵) ≼ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
257, 23, 24syl2anc 573 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
26 mapdom1 8281 . . . 4 (2𝑜𝐴 → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
2726ad2antrl 707 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
28 endomtr 8167 . . 3 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
293, 27, 28syl2anc 573 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
30 sbth 8236 . 2 (((𝐴𝑚 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
3125, 29, 30syl2anc 573 1 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  𝒫 cpw 4297   class class class wbr 4786   × cxp 5247  dom cdm 5249  Oncon0 5866  (class class class)co 6793  ωcom 7212  2𝑜c2o 7707  𝑚 cmap 8009  cen 8106  cdom 8107  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-oi 8571  df-card 8965
This theorem is referenced by:  alephexp1  9603  hauspwdom  21525
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