Theorem cenc 6182

Description: Cardinal exponentiation in terms of cardinality. Theorem XI.2.39 of [Rosser] p. 382. (Contributed by SF, 6-Mar-2015.)

Hypotheses

Ref	Expression
cenc.1	⊢ A ∈ V
cenc.2	⊢ B ∈ V

Assertion

Ref	Expression
cenc	⊢ ( Nc 𝒫1A ↑c Nc 𝒫1B) = Nc (A ↑m B)

Proof of Theorem cenc

Dummy variables a b g p t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnc 6126	. . . . . . . . 9 ⊢ (𝒫1p ∈ Nc 𝒫1A ↔ 𝒫1p ≈ 𝒫1A)
2		enpw1 6063	. . . . . . . . 9 ⊢ (p ≈ A ↔ 𝒫1p ≈ 𝒫1A)
3	1, 2	bitr4i 243	. . . . . . . 8 ⊢ (𝒫1p ∈ Nc 𝒫1A ↔ p ≈ A)
4		elnc 6126	. . . . . . . . 9 ⊢ (𝒫1t ∈ Nc 𝒫1B ↔ 𝒫1t ≈ 𝒫1B)
5		enpw1 6063	. . . . . . . . 9 ⊢ (t ≈ B ↔ 𝒫1t ≈ 𝒫1B)
6	4, 5	bitr4i 243	. . . . . . . 8 ⊢ (𝒫1t ∈ Nc 𝒫1B ↔ t ≈ B)
7		enmap1 6075	. . . . . . . . 9 ⊢ (p ≈ A → (p ↑m t) ≈ (A ↑m t))
8		enmap2 6069	. . . . . . . . 9 ⊢ (t ≈ B → (A ↑m t) ≈ (A ↑m B))
9		entr 6039	. . . . . . . . 9 ⊢ (((p ↑m t) ≈ (A ↑m t) ∧ (A ↑m t) ≈ (A ↑m B)) → (p ↑m t) ≈ (A ↑m B))
10	7, 8, 9	syl2an 463	. . . . . . . 8 ⊢ ((p ≈ A ∧ t ≈ B) → (p ↑m t) ≈ (A ↑m B))
11	3, 6, 10	syl2anb 465	. . . . . . 7 ⊢ ((𝒫1p ∈ Nc 𝒫1A ∧ 𝒫1t ∈ Nc 𝒫1B) → (p ↑m t) ≈ (A ↑m B))
12		entr 6039	. . . . . . . 8 ⊢ ((g ≈ (p ↑m t) ∧ (p ↑m t) ≈ (A ↑m B)) → g ≈ (A ↑m B))
13	12	ancoms 439	. . . . . . 7 ⊢ (((p ↑m t) ≈ (A ↑m B) ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
14	11, 13	sylan 457	. . . . . 6 ⊢ (((𝒫1p ∈ Nc 𝒫1A ∧ 𝒫1t ∈ Nc 𝒫1B) ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
15	14	3impa 1146	. . . . 5 ⊢ ((𝒫1p ∈ Nc 𝒫1A ∧ 𝒫1t ∈ Nc 𝒫1B ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
16	15	exlimivv 1635	. . . 4 ⊢ (∃p∃t(𝒫1p ∈ Nc 𝒫1A ∧ 𝒫1t ∈ Nc 𝒫1B ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
17		cenc.1	. . . . . . 7 ⊢ A ∈ V
18	17	pw1ex 4304	. . . . . 6 ⊢ 𝒫1A ∈ V
19	18	ncelncsi 6122	. . . . 5 ⊢ Nc 𝒫1A ∈ NC
20		cenc.2	. . . . . . 7 ⊢ B ∈ V
21	20	pw1ex 4304	. . . . . 6 ⊢ 𝒫1B ∈ V
22	21	ncelncsi 6122	. . . . 5 ⊢ Nc 𝒫1B ∈ NC
23		elce 6176	. . . . 5 ⊢ (( Nc 𝒫1A ∈ NC ∧ Nc 𝒫1B ∈ NC ) → (g ∈ ( Nc 𝒫1A ↑c Nc 𝒫1B) ↔ ∃p∃t(𝒫1p ∈ Nc 𝒫1A ∧ 𝒫1t ∈ Nc 𝒫1B ∧ g ≈ (p ↑m t))))
24	19, 22, 23	mp2an 653	. . . 4 ⊢ (g ∈ ( Nc 𝒫1A ↑c Nc 𝒫1B) ↔ ∃p∃t(𝒫1p ∈ Nc 𝒫1A ∧ 𝒫1t ∈ Nc 𝒫1B ∧ g ≈ (p ↑m t)))
25		elnc 6126	. . . 4 ⊢ (g ∈ Nc (A ↑m B) ↔ g ≈ (A ↑m B))
26	16, 24, 25	3imtr4i 257	. . 3 ⊢ (g ∈ ( Nc 𝒫1A ↑c Nc 𝒫1B) → g ∈ Nc (A ↑m B))
27	26	ssriv 3278	. 2 ⊢ ( Nc 𝒫1A ↑c Nc 𝒫1B) ⊆ Nc (A ↑m B)
28	18	ncid 6124	. . . . 5 ⊢ 𝒫1A ∈ Nc 𝒫1A
29	21	ncid 6124	. . . . 5 ⊢ 𝒫1B ∈ Nc 𝒫1B
30		pw1eq 4144	. . . . . . . . 9 ⊢ (a = A → 𝒫1a = 𝒫1A)
31	30	eleq1d 2419	. . . . . . . 8 ⊢ (a = A → (𝒫1a ∈ Nc 𝒫1A ↔ 𝒫1A ∈ Nc 𝒫1A))
32	31	adantr 451	. . . . . . 7 ⊢ ((a = A ∧ b = B) → (𝒫1a ∈ Nc 𝒫1A ↔ 𝒫1A ∈ Nc 𝒫1A))
33		pw1eq 4144	. . . . . . . . 9 ⊢ (b = B → 𝒫1b = 𝒫1B)
34	33	eleq1d 2419	. . . . . . . 8 ⊢ (b = B → (𝒫1b ∈ Nc 𝒫1B ↔ 𝒫1B ∈ Nc 𝒫1B))
35	34	adantl 452	. . . . . . 7 ⊢ ((a = A ∧ b = B) → (𝒫1b ∈ Nc 𝒫1B ↔ 𝒫1B ∈ Nc 𝒫1B))
36		oveq12 5533	. . . . . . . 8 ⊢ ((a = A ∧ b = B) → (a ↑m b) = (A ↑m B))
37	36	breq2d 4652	. . . . . . 7 ⊢ ((a = A ∧ b = B) → (g ≈ (a ↑m b) ↔ g ≈ (A ↑m B)))
38	32, 35, 37	3anbi123d 1252	. . . . . 6 ⊢ ((a = A ∧ b = B) → ((𝒫1a ∈ Nc 𝒫1A ∧ 𝒫1b ∈ Nc 𝒫1B ∧ g ≈ (a ↑m b)) ↔ (𝒫1A ∈ Nc 𝒫1A ∧ 𝒫1B ∈ Nc 𝒫1B ∧ g ≈ (A ↑m B))))
39	17, 20, 38	spc2ev 2948	. . . . 5 ⊢ ((𝒫1A ∈ Nc 𝒫1A ∧ 𝒫1B ∈ Nc 𝒫1B ∧ g ≈ (A ↑m B)) → ∃a∃b(𝒫1a ∈ Nc 𝒫1A ∧ 𝒫1b ∈ Nc 𝒫1B ∧ g ≈ (a ↑m b)))
40	28, 29, 39	mp3an12 1267	. . . 4 ⊢ (g ≈ (A ↑m B) → ∃a∃b(𝒫1a ∈ Nc 𝒫1A ∧ 𝒫1b ∈ Nc 𝒫1B ∧ g ≈ (a ↑m b)))
41		elce 6176	. . . . 5 ⊢ (( Nc 𝒫1A ∈ NC ∧ Nc 𝒫1B ∈ NC ) → (g ∈ ( Nc 𝒫1A ↑c Nc 𝒫1B) ↔ ∃a∃b(𝒫1a ∈ Nc 𝒫1A ∧ 𝒫1b ∈ Nc 𝒫1B ∧ g ≈ (a ↑m b))))
42	19, 22, 41	mp2an 653	. . . 4 ⊢ (g ∈ ( Nc 𝒫1A ↑c Nc 𝒫1B) ↔ ∃a∃b(𝒫1a ∈ Nc 𝒫1A ∧ 𝒫1b ∈ Nc 𝒫1B ∧ g ≈ (a ↑m b)))
43	40, 25, 42	3imtr4i 257	. . 3 ⊢ (g ∈ Nc (A ↑m B) → g ∈ ( Nc 𝒫1A ↑c Nc 𝒫1B))
44	43	ssriv 3278	. 2 ⊢ Nc (A ↑m B) ⊆ ( Nc 𝒫1A ↑c Nc 𝒫1B)
45	27, 44	eqssi 3289	1 ⊢ ( Nc 𝒫1A ↑c Nc 𝒫1B) = Nc (A ↑m B)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  𝒫₁cpw1 4136   class class class wbr 4640  (class class class)co 5526   ↑_m cmap 6000   ≈ cen 6029   NC cncs 6089   Nc cnc 6092   ↑_c cce 6097

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-ce 6107

This theorem is used by:  ce0nnulb  6183  ceclb  6184  ce0  6191  ce2  6193