Theorem cenc 6181

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 ℘1A ↑c Nc ℘1B) = 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 6125	. . . . . . . . 9 ⊢ (℘1p ∈ Nc ℘1A ↔ ℘1p ≈ ℘1A)
2		enpw1 6062	. . . . . . . . 9 ⊢ (p ≈ A ↔ ℘1p ≈ ℘1A)
3	1, 2	bitr4i 243	. . . . . . . 8 ⊢ (℘1p ∈ Nc ℘1A ↔ p ≈ A)
4		elnc 6125	. . . . . . . . 9 ⊢ (℘1t ∈ Nc ℘1B ↔ ℘1t ≈ ℘1B)
5		enpw1 6062	. . . . . . . . 9 ⊢ (t ≈ B ↔ ℘1t ≈ ℘1B)
6	4, 5	bitr4i 243	. . . . . . . 8 ⊢ (℘1t ∈ Nc ℘1B ↔ t ≈ B)
7		enmap1 6074	. . . . . . . . 9 ⊢ (p ≈ A → (p ↑m t) ≈ (A ↑m t))
8		enmap2 6068	. . . . . . . . 9 ⊢ (t ≈ B → (A ↑m t) ≈ (A ↑m B))
9		entr 6038	. . . . . . . . 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 ⊢ ((℘1p ∈ Nc ℘1A ∧ ℘1t ∈ Nc ℘1B) → (p ↑m t) ≈ (A ↑m B))
12		entr 6038	. . . . . . . 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 ⊢ (((℘1p ∈ Nc ℘1A ∧ ℘1t ∈ Nc ℘1B) ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
15	14	3impa 1146	. . . . 5 ⊢ ((℘1p ∈ Nc ℘1A ∧ ℘1t ∈ Nc ℘1B ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
16	15	exlimivv 1635	. . . 4 ⊢ (∃p∃t(℘1p ∈ Nc ℘1A ∧ ℘1t ∈ Nc ℘1B ∧ g ≈ (p ↑m t)) → g ≈ (A ↑m B))
17		cenc.1	. . . . . . 7 ⊢ A ∈ V
18	17	pw1ex 4303	. . . . . 6 ⊢ ℘1A ∈ V
19	18	ncelncsi 6121	. . . . 5 ⊢ Nc ℘1A ∈ NC
20		cenc.2	. . . . . . 7 ⊢ B ∈ V
21	20	pw1ex 4303	. . . . . 6 ⊢ ℘1B ∈ V
22	21	ncelncsi 6121	. . . . 5 ⊢ Nc ℘1B ∈ NC
23		elce 6175	. . . . 5 ⊢ (( Nc ℘1A ∈ NC ∧ Nc ℘1B ∈ NC ) → (g ∈ ( Nc ℘1A ↑c Nc ℘1B) ↔ ∃p∃t(℘1p ∈ Nc ℘1A ∧ ℘1t ∈ Nc ℘1B ∧ g ≈ (p ↑m t))))
24	19, 22, 23	mp2an 653	. . . 4 ⊢ (g ∈ ( Nc ℘1A ↑c Nc ℘1B) ↔ ∃p∃t(℘1p ∈ Nc ℘1A ∧ ℘1t ∈ Nc ℘1B ∧ g ≈ (p ↑m t)))
25		elnc 6125	. . . 4 ⊢ (g ∈ Nc (A ↑m B) ↔ g ≈ (A ↑m B))
26	16, 24, 25	3imtr4i 257	. . 3 ⊢ (g ∈ ( Nc ℘1A ↑c Nc ℘1B) → g ∈ Nc (A ↑m B))
27	26	ssriv 3277	. 2 ⊢ ( Nc ℘1A ↑c Nc ℘1B) ⊆ Nc (A ↑m B)
28	18	ncid 6123	. . . . 5 ⊢ ℘1A ∈ Nc ℘1A
29	21	ncid 6123	. . . . 5 ⊢ ℘1B ∈ Nc ℘1B
30		pw1eq 4143	. . . . . . . . 9 ⊢ (a = A → ℘1a = ℘1A)
31	30	eleq1d 2419	. . . . . . . 8 ⊢ (a = A → (℘1a ∈ Nc ℘1A ↔ ℘1A ∈ Nc ℘1A))
32	31	adantr 451	. . . . . . 7 ⊢ ((a = A ∧ b = B) → (℘1a ∈ Nc ℘1A ↔ ℘1A ∈ Nc ℘1A))
33		pw1eq 4143	. . . . . . . . 9 ⊢ (b = B → ℘1b = ℘1B)
34	33	eleq1d 2419	. . . . . . . 8 ⊢ (b = B → (℘1b ∈ Nc ℘1B ↔ ℘1B ∈ Nc ℘1B))
35	34	adantl 452	. . . . . . 7 ⊢ ((a = A ∧ b = B) → (℘1b ∈ Nc ℘1B ↔ ℘1B ∈ Nc ℘1B))
36		oveq12 5532	. . . . . . . 8 ⊢ ((a = A ∧ b = B) → (a ↑m b) = (A ↑m B))
37	36	breq2d 4651	. . . . . . 7 ⊢ ((a = A ∧ b = B) → (g ≈ (a ↑m b) ↔ g ≈ (A ↑m B)))
38	32, 35, 37	3anbi123d 1252	. . . . . 6 ⊢ ((a = A ∧ b = B) → ((℘1a ∈ Nc ℘1A ∧ ℘1b ∈ Nc ℘1B ∧ g ≈ (a ↑m b)) ↔ (℘1A ∈ Nc ℘1A ∧ ℘1B ∈ Nc ℘1B ∧ g ≈ (A ↑m B))))
39	17, 20, 38	spc2ev 2947	. . . . 5 ⊢ ((℘1A ∈ Nc ℘1A ∧ ℘1B ∈ Nc ℘1B ∧ g ≈ (A ↑m B)) → ∃a∃b(℘1a ∈ Nc ℘1A ∧ ℘1b ∈ Nc ℘1B ∧ g ≈ (a ↑m b)))
40	28, 29, 39	mp3an12 1267	. . . 4 ⊢ (g ≈ (A ↑m B) → ∃a∃b(℘1a ∈ Nc ℘1A ∧ ℘1b ∈ Nc ℘1B ∧ g ≈ (a ↑m b)))
41		elce 6175	. . . . 5 ⊢ (( Nc ℘1A ∈ NC ∧ Nc ℘1B ∈ NC ) → (g ∈ ( Nc ℘1A ↑c Nc ℘1B) ↔ ∃a∃b(℘1a ∈ Nc ℘1A ∧ ℘1b ∈ Nc ℘1B ∧ g ≈ (a ↑m b))))
42	19, 22, 41	mp2an 653	. . . 4 ⊢ (g ∈ ( Nc ℘1A ↑c Nc ℘1B) ↔ ∃a∃b(℘1a ∈ Nc ℘1A ∧ ℘1b ∈ Nc ℘1B ∧ g ≈ (a ↑m b)))
43	40, 25, 42	3imtr4i 257	. . 3 ⊢ (g ∈ Nc (A ↑m B) → g ∈ ( Nc ℘1A ↑c Nc ℘1B))
44	43	ssriv 3277	. 2 ⊢ Nc (A ↑m B) ⊆ ( Nc ℘1A ↑c Nc ℘1B)
45	27, 44	eqssi 3288	1 ⊢ ( Nc ℘1A ↑c Nc ℘1B) = Nc (A ↑m B)

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ℘₁cpw1 4135   class class class wbr 4639  (class class class)co 5525   ↑_m cmap 5999   ≈ cen 6028   NC cncs 6088   Nc cnc 6091   ↑_c cce 6096

This theorem was proved from 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-ce 6106

This theorem is referenced by:  ce0nnulb  6182  ceclb  6183  ce0  6190  ce2  6192