Theorem fnce 6177

Description: Functionhood statement for cardinal exponentiation. (Contributed by SF, 6-Mar-2015.)

Assertion

Ref	Expression
fnce	⊢ ↑c Fn ( NC × NC )

Proof of Theorem fnce

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

Step	Hyp	Ref	Expression
1		ovcelem1 6172	. . . . 5 ⊢ ((n ∈ NC ∧ m ∈ NC ) → {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))} ∈ V)
2		isset 2864	. . . . 5 ⊢ ({g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))} ∈ V ↔ ∃p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
3	1, 2	sylib 188	. . . 4 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ∃p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
4		moeq 3013	. . . . 5 ⊢ ∃*p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))}
5		eu5 2242	. . . . 5 ⊢ (∃!p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))} ↔ (∃p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))} ∧ ∃*p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))}))
6	4, 5	mpbiran2 885	. . . 4 ⊢ (∃!p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))} ↔ ∃p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
7	3, 6	sylibr 203	. . 3 ⊢ ((n ∈ NC ∧ m ∈ NC ) → ∃!p p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
8	7	fnoprab 5587	. 2 ⊢ {⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})} Fn {⟨n, m⟩ ∣ (n ∈ NC ∧ m ∈ NC )}
9		df-ce 6107	. . . . 5 ⊢ ↑c = (n ∈ NC , m ∈ NC ↦ {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
10		df-mpt2 5655	. . . . 5 ⊢ (n ∈ NC , m ∈ NC ↦ {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))}) = {⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})}
11	9, 10	eqtri 2373	. . . 4 ⊢ ↑c = {⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})}
12	11	fneq1i 5179	. . 3 ⊢ ( ↑c Fn ( NC × NC ) ↔ {⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})} Fn ( NC × NC ))
13		df-xp 4785	. . . 4 ⊢ ( NC × NC ) = {⟨n, m⟩ ∣ (n ∈ NC ∧ m ∈ NC )}
14	13	fneq2i 5180	. . 3 ⊢ ({⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})} Fn ( NC × NC ) ↔ {⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})} Fn {⟨n, m⟩ ∣ (n ∈ NC ∧ m ∈ NC )})
15	12, 14	bitri 240	. 2 ⊢ ( ↑c Fn ( NC × NC ) ↔ {⟨⟨n, m⟩, p⟩ ∣ ((n ∈ NC ∧ m ∈ NC ) ∧ p = {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})} Fn {⟨n, m⟩ ∣ (n ∈ NC ∧ m ∈ NC )})
16	8, 15	mpbir 200	1 ⊢ ↑c Fn ( NC × NC )

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  {cab 2339  Vcvv 2860  ℘₁cpw1 4136  {copab 4623   class class class wbr 4640   × cxp 4771   Fn wfn 4777  (class class class)co 5526  {coprab 5528   ↦ cmpt2 5654   ↑_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-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-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-map 6002  df-en 6030  df-ce 6107

This theorem is referenced by:  fce  6189