Theorem mappwen 8935

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 796	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ 𝒫 𝐵)
2		pw2eng 8066	. . . . . 6 ⊢ (𝐵 ∈ dom card → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
3	2	ad2antrr 762	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
4		domentr 8015	. . . . 5 ⊢ ((𝐴 ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵)) → 𝐴 ≼ (2𝑜 ↑𝑚 𝐵))
5	1, 3, 4	syl2anc 693	. . . 4 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝐴 ≼ (2𝑜 ↑𝑚 𝐵))
6		mapdom1 8125	. . . 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 7568	. . . . . . 7 ⊢ 2𝑜 ∈ On
9	8	a1i 11	. . . . . 6 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 2𝑜 ∈ On)
10		simpll 790	. . . . . 6 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝐵 ∈ dom card)
11		mapxpen 8126	. . . . . 6 ⊢ ((2𝑜 ∈ On ∧ 𝐵 ∈ dom card ∧ 𝐵 ∈ dom card) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐵)))
12	9, 10, 10, 11	syl3anc 1326	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐵)))
13	8	elexi 3213	. . . . . . 7 ⊢ 2𝑜 ∈ V
14	13	enref 7988	. . . . . 6 ⊢ 2𝑜 ≈ 2𝑜
15		infxpidm2 8840	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ ω ≼ 𝐵) → (𝐵 × 𝐵) ≈ 𝐵)
16	15	adantr 481	. . . . . 6 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐵 × 𝐵) ≈ 𝐵)
17		mapen 8124	. . . . . 6 ⊢ ((2𝑜 ≈ 2𝑜 ∧ (𝐵 × 𝐵) ≈ 𝐵) → (2𝑜 ↑𝑚 (𝐵 × 𝐵)) ≈ (2𝑜 ↑𝑚 𝐵))
18	14, 16, 17	sylancr 695	. . . . 5 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (2𝑜 ↑𝑚 (𝐵 × 𝐵)) ≈ (2𝑜 ↑𝑚 𝐵))
19		entr 8008	. . . . 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 693	. . . 4 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ (2𝑜 ↑𝑚 𝐵))
21	3	ensymd 8007	. . . 4 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵)
22		entr 8008	. . . 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 693	. . 3 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐵) ≈ 𝒫 𝐵)
24		domentr 8015	. . 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 693	. 2 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑𝑚 𝐵) ≼ 𝒫 𝐵)
26		mapdom1 8125	. . . 4 ⊢ (2𝑜 ≼ 𝐴 → (2𝑜 ↑𝑚 𝐵) ≼ (𝐴 ↑𝑚 𝐵))
27	26	ad2antrl 764	. . 3 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (2𝑜 ↑𝑚 𝐵) ≼ (𝐴 ↑𝑚 𝐵))
28		endomtr 8014	. . 3 ⊢ ((𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵) ∧ (2𝑜 ↑𝑚 𝐵) ≼ (𝐴 ↑𝑚 𝐵)) → 𝒫 𝐵 ≼ (𝐴 ↑𝑚 𝐵))
29	3, 27, 28	syl2anc 693	. 2 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → 𝒫 𝐵 ≼ (𝐴 ↑𝑚 𝐵))
30		sbth 8080	. 2 ⊢ (((𝐴 ↑𝑚 𝐵) ≼ 𝒫 𝐵 ∧ 𝒫 𝐵 ≼ (𝐴 ↑𝑚 𝐵)) → (𝐴 ↑𝑚 𝐵) ≈ 𝒫 𝐵)
31	25, 29, 30	syl2anc 693	1 ⊢ (((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜 ≼ 𝐴 ∧ 𝐴 ≼ 𝒫 𝐵)) → (𝐴 ↑𝑚 𝐵) ≈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  𝒫 cpw 4158   class class class wbr 4653   × cxp 5112  dom cdm 5114  Oncon0 5723  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   ↑_𝑚 cmap 7857   ≈ cen 7952   ≼ cdom 7953  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765

This theorem is referenced by:  alephexp1  9401  hauspwdom  21304