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Theorem fnce 6176
 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.
StepHypRef Expression
1 ovcelem1 6171 . . . . 5 ((n NC m NC ) → {g ab(1a n 1b m g ≈ (am b))} V)
2 isset 2863 . . . . 5 ({g ab(1a n 1b m g ≈ (am b))} V ↔ p p = {g ab(1a n 1b m g ≈ (am b))})
31, 2sylib 188 . . . 4 ((n NC m NC ) → p p = {g ab(1a n 1b m g ≈ (am b))})
4 moeq 3012 . . . . 5 ∃*p p = {g ab(1a n 1b m g ≈ (am b))}
5 eu5 2242 . . . . 5 (∃!p p = {g ab(1a n 1b m g ≈ (am b))} ↔ (p p = {g ab(1a n 1b m g ≈ (am b))} ∃*p p = {g ab(1a n 1b m g ≈ (am b))}))
64, 5mpbiran2 885 . . . 4 (∃!p p = {g ab(1a n 1b m g ≈ (am b))} ↔ p p = {g ab(1a n 1b m g ≈ (am b))})
73, 6sylibr 203 . . 3 ((n NC m NC ) → ∃!p p = {g ab(1a n 1b m g ≈ (am b))})
87fnoprab 5586 . 2 {n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})} Fn {n, m (n NC m NC )}
9 df-ce 6106 . . . . 5 c = (n NC , m NC {g ab(1a n 1b m g ≈ (am b))})
10 df-mpt2 5654 . . . . 5 (n NC , m NC {g ab(1a n 1b m g ≈ (am b))}) = {n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})}
119, 10eqtri 2373 . . . 4 c = {n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})}
1211fneq1i 5178 . . 3 ( ↑c Fn ( NC × NC ) ↔ {n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})} Fn ( NC × NC ))
13 df-xp 4784 . . . 4 ( NC × NC ) = {n, m (n NC m NC )}
1413fneq2i 5179 . . 3 ({n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})} Fn ( NC × NC ) ↔ {n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})} Fn {n, m (n NC m NC )})
1512, 14bitri 240 . 2 ( ↑c Fn ( NC × NC ) ↔ {n, m, p ((n NC m NC ) p = {g ab(1a n 1b m g ≈ (am b))})} Fn {n, m (n NC m NC )})
168, 15mpbir 200 1 c Fn ( NC × NC )
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  {cab 2339  Vcvv 2859  ℘1cpw1 4135  {copab 4622   class class class wbr 4639   × cxp 4770   Fn wfn 4776  (class class class)co 5525  {coprab 5527   ↦ cmpt2 5653   ↑m cmap 5999   ≈ cen 6028   NC cncs 6088   ↑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-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-map 6001  df-en 6029  df-ce 6106 This theorem is referenced by:  fce  6188
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