MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mappwen Structured version   Visualization version   GIF version

Theorem mappwen 8818
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 792 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2 pw2eng 7951 . . . . . 6 (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
32ad2antrr 758 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
4 domentr 7901 . . . . 5 ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
51, 3, 4syl2anc 691 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2𝑜𝑚 𝐵))
6 mapdom1 8010 . . . 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 7455 . . . . . . 7 2𝑜 ∈ On
98a1i 11 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 2𝑜 ∈ On)
10 simpll 786 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
11 mapxpen 8011 . . . . . 6 ((2𝑜 ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
129, 10, 10, 11syl3anc 1318 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)))
138elexi 3186 . . . . . . 7 2𝑜 ∈ V
1413enref 7874 . . . . . 6 2𝑜 ≈ 2𝑜
15 infxpidm2 8723 . . . . . . 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 8009 . . . . . 6 ((2𝑜 ≈ 2𝑜 ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
1814, 16, 17sylancr 694 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵))
19 entr 7894 . . . . 5 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐵)) ∧ (2𝑜𝑚 (𝐵 × 𝐵)) ≈ (2𝑜𝑚 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
2012, 18, 19syl2anc 691 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵))
213ensymd 7893 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵)
22 entr 7894 . . . 4 ((((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
2320, 21, 22syl2anc 691 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
24 domentr 7901 . . 3 (((𝐴𝑚 𝐵) ≼ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
257, 23, 24syl2anc 691 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≼ 𝒫 𝐵)
26 mapdom1 8010 . . . 4 (2𝑜𝐴 → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
2726ad2antrl 760 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵))
28 endomtr 7900 . . 3 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≼ (𝐴𝑚 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
293, 27, 28syl2anc 691 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴𝑚 𝐵))
30 sbth 7965 . 2 (((𝐴𝑚 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
3125, 29, 30syl2anc 691 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 1977  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  dom cdm 5038  Oncon0 5640  (class class class)co 6549  ωcom 6957  2𝑜c2o 7441  𝑚 cmap 7744  cen 7838  cdom 7839  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648
This theorem is referenced by:  alephexp1  9280  hauspwdom  21114
  Copyright terms: Public domain W3C validator