Theorem mappwen 9221

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) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑𝑚 𝐵) ≈ 𝒫 𝐵)

Proof of Theorem mappwen

Step	Hyp	Ref	Expression
1		simprr 790	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2		pw2eng 8308	. . . . . 6 ⊢ (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
3	2	ad2antrr 718	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
4		domentr 8254	. . . . 5 ⊢ ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵)) → 𝐴 ≼ (2𝑜 ↑𝑚 𝐵))
5	1, 3, 4	syl2anc 580	. . . 4 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2𝑜 ↑𝑚 𝐵))
6		mapdom1 8367	. . . 4 ⊢ (𝐴 ≼ (2𝑜 ↑𝑚 𝐵) → (𝐴 ↑𝑚 𝐵) ≼ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵))
7	5, 6	syl 17	. . 3 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑𝑚 𝐵) ≼ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵))
8		2on 7808	. . . . . . 7 ⊢ 2𝑜 ∈ On
9	8	a1i 11	. . . . . 6 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 2𝑜 ∈ On)
10		simpll 784	. . . . . 6 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
11		mapxpen 8368	. . . . . 6 ⊢ ((2𝑜 ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐵)))
12	9, 10, 10, 11	syl3anc 1491	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐵)))
13	8	elexi 3401	. . . . . . 7 ⊢ 2𝑜 ∈ V
14	13	enref 8228	. . . . . 6 ⊢ 2𝑜 ≈ 2𝑜
15		infxpidm2 9126	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵) → (𝐵 × 𝐵) ≈ 𝐵)
16	15	adantr 473	. . . . . 6 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐵 × 𝐵) ≈ 𝐵)
17		mapen 8366	. . . . . 6 ⊢ ((2𝑜 ≈ 2𝑜 ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2𝑜 ↑𝑚 (𝐵 × 𝐵)) ≈ (2𝑜 ↑𝑚 𝐵))
18	14, 16, 17	sylancr 582	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (2𝑜 ↑𝑚 (𝐵 × 𝐵)) ≈ (2𝑜 ↑𝑚 𝐵))
19		entr 8247	. . . . 5 ⊢ ((((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐵)) ∧ (2𝑜 ↑𝑚 (𝐵 × 𝐵)) ≈ (2𝑜 ↑𝑚 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 𝐵))
20	12, 18, 19	syl2anc 580	. . . 4 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 𝐵))
21	3	ensymd 8246	. . . 4 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵)
22		entr 8247	. . . 4 ⊢ ((((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 𝐵) ∧ (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
23	20, 21, 22	syl2anc 580	. . 3 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
24		domentr 8254	. . 3 ⊢ (((𝐴 ↑𝑚 𝐵) ≼ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ∧ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (𝐴 ↑𝑚 𝐵) ≼ 𝒫 𝐵)
25	7, 23, 24	syl2anc 580	. 2 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑𝑚 𝐵) ≼ 𝒫 𝐵)
26		mapdom1 8367	. . . 4 ⊢ (2𝑜 ≼ 𝐴 → (2𝑜 ↑𝑚 𝐵) ≼ (𝐴 ↑𝑚 𝐵))
27	26	ad2antrl 720	. . 3 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (2𝑜 ↑𝑚 𝐵) ≼ (𝐴 ↑𝑚 𝐵))
28		endomtr 8253	. . 3 ⊢ ((𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵) ∧ (2𝑜 ↑𝑚 𝐵) ≼ (𝐴 ↑𝑚 𝐵)) → 𝒫 𝐵 ≼ (𝐴 ↑𝑚 𝐵))
29	3, 27, 28	syl2anc 580	. 2 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴 ↑𝑚 𝐵))
30		sbth 8322	. 2 ⊢ (((𝐴 ↑𝑚 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴 ↑𝑚 𝐵)) → (𝐴 ↑𝑚 𝐵) ≈ 𝒫 𝐵)
31	25, 29, 30	syl2anc 580	1 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑𝑚 𝐵) ≈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 385   ∈ wcel 2157  𝒫 cpw 4349   class class class wbr 4843   × cxp 5310  dom cdm 5312  Oncon0 5941  (class class class)co 6878  ωcom 7299  2_𝑜c2o 7793   ↑_𝑚 cmap 8095   ≈ cen 8192   ≼ cdom 8193  cardccrd 9047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-oi 8657  df-card 9051

This theorem is referenced by:  alephexp1  9689  hauspwdom  21633