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

Theorem mappwen 9530
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 771 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2 pw2eng 8615 . . . . . 6 (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2om 𝐵))
32ad2antrr 724 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2om 𝐵))
4 domentr 8560 . . . . 5 ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2om 𝐵)) → 𝐴 ≼ (2om 𝐵))
51, 3, 4syl2anc 586 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2om 𝐵))
6 mapdom1 8674 . . . 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 8103 . . . . . 6 2o ∈ On
9 simpll 765 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
10 mapxpen 8675 . . . . . 6 ((2o ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2om 𝐵) ↑m 𝐵) ≈ (2om (𝐵 × 𝐵)))
118, 9, 9, 10mp3an2i 1459 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → ((2om 𝐵) ↑m 𝐵) ≈ (2om (𝐵 × 𝐵)))
128elexi 3512 . . . . . . 7 2o ∈ V
1312enref 8534 . . . . . 6 2o ≈ 2o
14 infxpidm2 9435 . . . . . . 7 ((𝐵 ∈ dom card ∧ ω ≼ 𝐵) → (𝐵 × 𝐵) ≈ 𝐵)
1514adantr 483 . . . . . 6 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐵 × 𝐵) ≈ 𝐵)
16 mapen 8673 . . . . . 6 ((2o ≈ 2o ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2om (𝐵 × 𝐵)) ≈ (2om 𝐵))
1713, 15, 16sylancr 589 . . . . 5 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → (2om (𝐵 × 𝐵)) ≈ (2om 𝐵))
18 entr 8553 . . . . 5 ((((2om 𝐵) ↑m 𝐵) ≈ (2om (𝐵 × 𝐵)) ∧ (2om (𝐵 × 𝐵)) ≈ (2om 𝐵)) → ((2om 𝐵) ↑m 𝐵) ≈ (2om 𝐵))
1911, 17, 18syl2anc 586 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → ((2om 𝐵) ↑m 𝐵) ≈ (2om 𝐵))
203ensymd 8552 . . . 4 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → (2om 𝐵) ≈ 𝒫 𝐵)
21 entr 8553 . . . 4 ((((2om 𝐵) ↑m 𝐵) ≈ (2om 𝐵) ∧ (2om 𝐵) ≈ 𝒫 𝐵) → ((2om 𝐵) ↑m 𝐵) ≈ 𝒫 𝐵)
2219, 20, 21syl2anc 586 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → ((2om 𝐵) ↑m 𝐵) ≈ 𝒫 𝐵)
23 domentr 8560 . . 3 (((𝐴m 𝐵) ≼ ((2om 𝐵) ↑m 𝐵) ∧ ((2om 𝐵) ↑m 𝐵) ≈ 𝒫 𝐵) → (𝐴m 𝐵) ≼ 𝒫 𝐵)
247, 22, 23syl2anc 586 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴m 𝐵) ≼ 𝒫 𝐵)
25 mapdom1 8674 . . . 4 (2o𝐴 → (2om 𝐵) ≼ (𝐴m 𝐵))
2625ad2antrl 726 . . 3 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → (2om 𝐵) ≼ (𝐴m 𝐵))
27 endomtr 8559 . . 3 ((𝒫 𝐵 ≈ (2om 𝐵) ∧ (2om 𝐵) ≼ (𝐴m 𝐵)) → 𝒫 𝐵 ≼ (𝐴m 𝐵))
283, 26, 27syl2anc 586 . 2 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴m 𝐵))
29 sbth 8629 . 2 (((𝐴m 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴m 𝐵)) → (𝐴m 𝐵) ≈ 𝒫 𝐵)
3024, 28, 29syl2anc 586 1 (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2o𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴m 𝐵) ≈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2107  𝒫 cpw 4537   class class class wbr 5057   × cxp 5546  dom cdm 5548  Oncon0 6184  (class class class)co 7148  ωcom 7572  2oc2o 8088  m cmap 8398  cen 8498  cdom 8499  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-card 9360
This theorem is referenced by:  alephexp1  9993  hauspwdom  22101
  Copyright terms: Public domain W3C validator