Theorem fce 6189

Description: Full functionhood statement for cardinal exponentiation. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
fce	⊢ ↑c :( NC × NC )–→( NC ∪ {∅})

Proof of Theorem fce

Dummy variables m n p x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnce 6177	. 2 ⊢ ↑c Fn ( NC × NC )
2		df-ne 2519	. . . . . . 7 ⊢ ((n ↑c m) ≠ ∅ ↔ ¬ (n ↑c m) = ∅)
3		n0 3560	. . . . . . . . 9 ⊢ ((n ↑c m) ≠ ∅ ↔ ∃p p ∈ (n ↑c m))
4		elce 6176	. . . . . . . . . . 11 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (p ∈ (n ↑c m) ↔ ∃x∃y(℘1x ∈ n ∧ ℘1y ∈ m ∧ p ≈ (x ↑m y))))
5		3simpa 952	. . . . . . . . . . . . 13 ⊢ ((℘1x ∈ n ∧ ℘1y ∈ m ∧ p ≈ (x ↑m y)) → (℘1x ∈ n ∧ ℘1y ∈ m))
6		ce0nnuli 6179	. . . . . . . . . . . . . . 15 ⊢ ((n ∈ NC ∧ ℘1x ∈ n) → (n ↑c 0c) ≠ ∅)
7	6	ex 423	. . . . . . . . . . . . . 14 ⊢ (n ∈ NC → (℘1x ∈ n → (n ↑c 0c) ≠ ∅))
8		ce0nnuli 6179	. . . . . . . . . . . . . . 15 ⊢ ((m ∈ NC ∧ ℘1y ∈ m) → (m ↑c 0c) ≠ ∅)
9	8	ex 423	. . . . . . . . . . . . . 14 ⊢ (m ∈ NC → (℘1y ∈ m → (m ↑c 0c) ≠ ∅))
10	7, 9	im2anan9 808	. . . . . . . . . . . . 13 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ((℘1x ∈ n ∧ ℘1y ∈ m) → ((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅)))
11	5, 10	syl5 28	. . . . . . . . . . . 12 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ((℘1x ∈ n ∧ ℘1y ∈ m ∧ p ≈ (x ↑m y)) → ((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅)))
12	11	exlimdvv 1637	. . . . . . . . . . 11 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (∃x∃y(℘1x ∈ n ∧ ℘1y ∈ m ∧ p ≈ (x ↑m y)) → ((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅)))
13	4, 12	sylbid 206	. . . . . . . . . 10 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (p ∈ (n ↑c m) → ((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅)))
14	13	exlimdv 1636	. . . . . . . . 9 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (∃p p ∈ (n ↑c m) → ((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅)))
15	3, 14	syl5bi 208	. . . . . . . 8 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ((n ↑c m) ≠ ∅ → ((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅)))
16		ceclb 6184	. . . . . . . 8 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (((n ↑c 0c) ≠ ∅ ∧ (m ↑c 0c) ≠ ∅) ↔ (n ↑c m) ∈ NC ))
17	15, 16	sylibd 205	. . . . . . 7 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ((n ↑c m) ≠ ∅ → (n ↑c m) ∈ NC ))
18	2, 17	syl5bir 209	. . . . . 6 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (¬ (n ↑c m) = ∅ → (n ↑c m) ∈ NC ))
19		elun 3221	. . . . . . 7 ⊢ ((n ↑c m) ∈ ( NC ∪ {∅}) ↔ ((n ↑c m) ∈ NC ∨ (n ↑c m) ∈ {∅}))
20		ovex 5552	. . . . . . . . . 10 ⊢ (n ↑c m) ∈ V
21	20	elsnc 3757	. . . . . . . . 9 ⊢ ((n ↑c m) ∈ {∅} ↔ (n ↑c m) = ∅)
22	21	orbi2i 505	. . . . . . . 8 ⊢ (((n ↑c m) ∈ NC ∨ (n ↑c m) ∈ {∅}) ↔ ((n ↑c m) ∈ NC ∨ (n ↑c m) = ∅))
23		orcom 376	. . . . . . . . 9 ⊢ (((n ↑c m) ∈ NC ∨ (n ↑c m) = ∅) ↔ ((n ↑c m) = ∅ ∨ (n ↑c m) ∈ NC ))
24		df-or 359	. . . . . . . . 9 ⊢ (((n ↑c m) = ∅ ∨ (n ↑c m) ∈ NC ) ↔ (¬ (n ↑c m) = ∅ → (n ↑c m) ∈ NC ))
25	23, 24	bitri 240	. . . . . . . 8 ⊢ (((n ↑c m) ∈ NC ∨ (n ↑c m) = ∅) ↔ (¬ (n ↑c m) = ∅ → (n ↑c m) ∈ NC ))
26	22, 25	bitri 240	. . . . . . 7 ⊢ (((n ↑c m) ∈ NC ∨ (n ↑c m) ∈ {∅}) ↔ (¬ (n ↑c m) = ∅ → (n ↑c m) ∈ NC ))
27	19, 26	bitri 240	. . . . . 6 ⊢ ((n ↑c m) ∈ ( NC ∪ {∅}) ↔ (¬ (n ↑c m) = ∅ → (n ↑c m) ∈ NC ))
28	18, 27	sylibr 203	. . . . 5 ⊢ ((n ∈ NC ∧ m ∈ NC ) → (n ↑c m) ∈ ( NC ∪ {∅}))
29	28	rgen2a 2681	. . . 4 ⊢ ∀n ∈ NC ∀m ∈ NC (n ↑c m) ∈ ( NC ∪ {∅})
30		fveq2 5329	. . . . . . 7 ⊢ (p = ⟨n, m⟩ → ( ↑c ‘p) = ( ↑c ‘⟨n, m⟩))
31		df-ov 5527	. . . . . . 7 ⊢ (n ↑c m) = ( ↑c ‘⟨n, m⟩)
32	30, 31	syl6eqr 2403	. . . . . 6 ⊢ (p = ⟨n, m⟩ → ( ↑c ‘p) = (n ↑c m))
33	32	eleq1d 2419	. . . . 5 ⊢ (p = ⟨n, m⟩ → (( ↑c ‘p) ∈ ( NC ∪ {∅}) ↔ (n ↑c m) ∈ ( NC ∪ {∅})))
34	33	ralxp 4826	. . . 4 ⊢ (∀p ∈ ( NC × NC )( ↑c ‘p) ∈ ( NC ∪ {∅}) ↔ ∀n ∈ NC ∀m ∈ NC (n ↑c m) ∈ ( NC ∪ {∅}))
35	29, 34	mpbir 200	. . 3 ⊢ ∀p ∈ ( NC × NC )( ↑c ‘p) ∈ ( NC ∪ {∅})
36		fnfvrnss 5430	. . 3 ⊢ (( ↑c Fn ( NC × NC ) ∧ ∀p ∈ ( NC × NC )( ↑c ‘p) ∈ ( NC ∪ {∅})) → ran ↑c ⊆ ( NC ∪ {∅}))
37	1, 35, 36	mp2an 653	. 2 ⊢ ran ↑c ⊆ ( NC ∪ {∅})
38		df-f 4792	. 2 ⊢ ( ↑c :( NC × NC )–→( NC ∪ {∅}) ↔ ( ↑c Fn ( NC × NC ) ∧ ran ↑c ⊆ ( NC ∪ {∅})))
39	1, 37, 38	mpbir2an 886	1 ⊢ ↑c :( NC × NC )–→( NC ∪ {∅})

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615   ∪ cun 3208   ⊆ wss 3258  ∅c0 3551  {csn 3738  ℘₁cpw1 4136  0_cc0c 4375  ⟨cop 4562   class class class wbr 4640   × cxp 4771  ran crn 4774   Fn wfn 4777  –→wf 4778   ‘cfv 4782  (class class class)co 5526   ↑_m cmap 6000   ≈ cen 6029   NC cncs 6089   ↑_c cce 6097

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 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 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 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-csb 3138  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-iun 3972  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 referenced by: (None)