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Theorem fnfullfun 5858
 Description: The full function operator yields a function over V. (Contributed by SF, 9-Mar-2015.)
Assertion
Ref Expression
fnfullfun FullFun F Fn V

Proof of Theorem fnfullfun
StepHypRef Expression
1 fnfullfunlem2 5857 . . . . 5 Fun (( I F) ( ∼ I F))
2 funfn 5136 . . . . 5 (Fun (( I F) ( ∼ I F)) ↔ (( I F) ( ∼ I F)) Fn dom (( I F) ( ∼ I F)))
31, 2mpbi 199 . . . 4 (( I F) ( ∼ I F)) Fn dom (( I F) ( ∼ I F))
4 0ex 4110 . . . . 5 V
5 fnconstg 5252 . . . . 5 ( V → ( ∼ dom (( I F) ( ∼ I F)) × {}) Fn ∼ dom (( I F) ( ∼ I F)))
64, 5ax-mp 5 . . . 4 ( ∼ dom (( I F) ( ∼ I F)) × {}) Fn ∼ dom (( I F) ( ∼ I F))
73, 6pm3.2i 441 . . 3 ((( I F) ( ∼ I F)) Fn dom (( I F) ( ∼ I F)) ( ∼ dom (( I F) ( ∼ I F)) × {}) Fn ∼ dom (( I F) ( ∼ I F)))
8 incompl 4073 . . 3 (dom (( I F) ( ∼ I F)) ∩ ∼ dom (( I F) ( ∼ I F))) =
9 fnun 5189 . . 3 ((((( I F) ( ∼ I F)) Fn dom (( I F) ( ∼ I F)) ( ∼ dom (( I F) ( ∼ I F)) × {}) Fn ∼ dom (( I F) ( ∼ I F))) (dom (( I F) ( ∼ I F)) ∩ ∼ dom (( I F) ( ∼ I F))) = ) → ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F))))
107, 8, 9mp2an 653 . 2 ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F)))
11 df-fullfun 5768 . . 3 FullFun F = ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {}))
12 uncompl 4074 . . . 4 (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F))) = V
1312eqcomi 2357 . . 3 V = (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F)))
14 fneq1 5173 . . . 4 ( FullFun F = ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) → ( FullFun F Fn V ↔ ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn V))
15 fneq2 5174 . . . 4 (V = (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F))) → (((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn V ↔ ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F)))))
1614, 15sylan9bb 680 . . 3 (( FullFun F = ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) V = (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F)))) → ( FullFun F Fn V ↔ ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F)))))
1711, 13, 16mp2an 653 . 2 ( FullFun F Fn V ↔ ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) Fn (dom (( I F) ( ∼ I F)) ∪ ∼ dom (( I F) ( ∼ I F))))
1810, 17mpbir 200 1 FullFun F Fn V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737   ∘ ccom 4721   I cid 4763   × cxp 4770  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   FullFun cfullfun 5767 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-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fullfun 5768 This theorem is referenced by:  brfullfung  5865
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