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Theorem fce 6188
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.
StepHypRef Expression
1 fnce 6176 . 2 c Fn ( NC × NC )
2 df-ne 2518 . . . . . . 7 ((nc m) ≠ ↔ ¬ (nc m) = )
3 n0 3559 . . . . . . . . 9 ((nc m) ≠ p p (nc m))
4 elce 6175 . . . . . . . . . . 11 ((n NC m NC ) → (p (nc m) ↔ xy(1x n 1y m p ≈ (xm y))))
5 3simpa 952 . . . . . . . . . . . . 13 ((1x n 1y m p ≈ (xm y)) → (1x n 1y m))
6 ce0nnuli 6178 . . . . . . . . . . . . . . 15 ((n NC 1x n) → (nc 0c) ≠ )
76ex 423 . . . . . . . . . . . . . 14 (n NC → (1x n → (nc 0c) ≠ ))
8 ce0nnuli 6178 . . . . . . . . . . . . . . 15 ((m NC 1y m) → (mc 0c) ≠ )
98ex 423 . . . . . . . . . . . . . 14 (m NC → (1y m → (mc 0c) ≠ ))
107, 9im2anan9 808 . . . . . . . . . . . . 13 ((n NC m NC ) → ((1x n 1y m) → ((nc 0c) ≠ (mc 0c) ≠ )))
115, 10syl5 28 . . . . . . . . . . . 12 ((n NC m NC ) → ((1x n 1y m p ≈ (xm y)) → ((nc 0c) ≠ (mc 0c) ≠ )))
1211exlimdvv 1637 . . . . . . . . . . 11 ((n NC m NC ) → (xy(1x n 1y m p ≈ (xm y)) → ((nc 0c) ≠ (mc 0c) ≠ )))
134, 12sylbid 206 . . . . . . . . . 10 ((n NC m NC ) → (p (nc m) → ((nc 0c) ≠ (mc 0c) ≠ )))
1413exlimdv 1636 . . . . . . . . 9 ((n NC m NC ) → (p p (nc m) → ((nc 0c) ≠ (mc 0c) ≠ )))
153, 14syl5bi 208 . . . . . . . 8 ((n NC m NC ) → ((nc m) ≠ → ((nc 0c) ≠ (mc 0c) ≠ )))
16 ceclb 6183 . . . . . . . 8 ((n NC m NC ) → (((nc 0c) ≠ (mc 0c) ≠ ) ↔ (nc m) NC ))
1715, 16sylibd 205 . . . . . . 7 ((n NC m NC ) → ((nc m) ≠ → (nc m) NC ))
182, 17syl5bir 209 . . . . . 6 ((n NC m NC ) → (¬ (nc m) = → (nc m) NC ))
19 elun 3220 . . . . . . 7 ((nc m) ( NC ∪ {}) ↔ ((nc m) NC (nc m) {}))
20 ovex 5551 . . . . . . . . . 10 (nc m) V
2120elsnc 3756 . . . . . . . . 9 ((nc m) {} ↔ (nc m) = )
2221orbi2i 505 . . . . . . . 8 (((nc m) NC (nc m) {}) ↔ ((nc m) NC (nc m) = ))
23 orcom 376 . . . . . . . . 9 (((nc m) NC (nc m) = ) ↔ ((nc m) = (nc m) NC ))
24 df-or 359 . . . . . . . . 9 (((nc m) = (nc m) NC ) ↔ (¬ (nc m) = → (nc m) NC ))
2523, 24bitri 240 . . . . . . . 8 (((nc m) NC (nc m) = ) ↔ (¬ (nc m) = → (nc m) NC ))
2622, 25bitri 240 . . . . . . 7 (((nc m) NC (nc m) {}) ↔ (¬ (nc m) = → (nc m) NC ))
2719, 26bitri 240 . . . . . 6 ((nc m) ( NC ∪ {}) ↔ (¬ (nc m) = → (nc m) NC ))
2818, 27sylibr 203 . . . . 5 ((n NC m NC ) → (nc m) ( NC ∪ {}))
2928rgen2a 2680 . . . 4 n NC m NC (nc m) ( NC ∪ {})
30 fveq2 5328 . . . . . . 7 (p = n, m → ( ↑cp) = ( ↑cn, m))
31 df-ov 5526 . . . . . . 7 (nc m) = ( ↑cn, m)
3230, 31syl6eqr 2403 . . . . . 6 (p = n, m → ( ↑cp) = (nc m))
3332eleq1d 2419 . . . . 5 (p = n, m → (( ↑cp) ( NC ∪ {}) ↔ (nc m) ( NC ∪ {})))
3433ralxp 4825 . . . 4 (p ( NC × NC )( ↑cp) ( NC ∪ {}) ↔ n NC m NC (nc m) ( NC ∪ {}))
3529, 34mpbir 200 . . 3 p ( NC × NC )( ↑cp) ( NC ∪ {})
36 fnfvrnss 5429 . . 3 (( ↑c Fn ( NC × NC ) p ( NC × NC )( ↑cp) ( NC ∪ {})) → ran ↑c ( NC ∪ {}))
371, 35, 36mp2an 653 . 2 ran ↑c ( NC ∪ {})
38 df-f 4791 . 2 ( ↑c :( NC × NC )–→( NC ∪ {}) ↔ ( ↑c Fn ( NC × NC ) ran ↑c ( NC ∪ {})))
391, 37, 38mpbir2an 886 1 c :( NC × NC )–→( NC ∪ {})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wo 357   wa 358   w3a 934  wex 1541   = wceq 1642   wcel 1710  wne 2516  wral 2614  cun 3207   wss 3257  c0 3550  {csn 3737  1cpw1 4135  0cc0c 4374  cop 4561   class class class wbr 4639   × cxp 4770  ran crn 4773   Fn wfn 4776  –→wf 4777  cfv 4781  (class class class)co 5525  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-csb 3137  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-iun 3971  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: (None)
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