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Theorem mappwen 8795
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 791 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2 pw2eng 7928 . . . . . 6 (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
32ad2antrr 757 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
4 domentr 7878 . . . . 5 ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
51, 3, 4syl2anc 690 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
6 mapdom1 7987 . . . 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 7432 . . . . . . 7 2𝑜 ∈ On
98a1i 11 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 2𝑜 ∈ On)
10 simpll 785 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
11 mapxpen 7988 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
129, 10, 10, 11syl3anc 1317 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
138elexi 3185 . . . . . . 7 2𝑜 ∈ V
1413enref 7851 . . . . . 6 2𝑜 ≈ 2𝑜
15 infxpidm2 8700 . . . . . . 7 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵) → (𝐵 × 𝐵) ≈ 𝐵)
1615adantr 479 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐵 × 𝐵) ≈ 𝐵)
17 mapen 7986 . . . . . 6 ((2𝑜 ≈ 2𝑜 ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
1814, 16, 17sylancr 693 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
19 entr 7871 . . . . 5 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)) ∧ (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
2012, 18, 19syl2anc 690 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
213ensymd 7870 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵)
22 entr 7871 . . . 4 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
2320, 21, 22syl2anc 690 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
24 domentr 7878 . . 3 (((𝐴𝑚 𝐵) ≼ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
257, 23, 24syl2anc 690 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
26 mapdom1 7987 . . . 4 (2𝑜𝐴 → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
2726ad2antrl 759 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
28 endomtr 7877 . . 3 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
293, 27, 28syl2anc 690 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
30 sbth 7942 . 2 (((𝐴𝑚 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
3125, 29, 30syl2anc 690 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 1976  𝒫 cpw 4107   class class class wbr 4577   × cxp 5026  dom cdm 5028  Oncon0 5626  (class class class)co 6527  ωcom 6934  2𝑜c2o 7418  𝑚 cmap 7721  cen 7815  cdom 7816  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-oi 8275  df-card 8625
This theorem is referenced by:  alephexp1  9257  hauspwdom  21056
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