Theorem fnfullfun 5859

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

Step	Hyp	Ref	Expression
1		fnfullfunlem2 5858	. . . . 5 ⊢ Fun (( I ∘ F) ∖ ( ∼ I ∘ F))
2		funfn 5137	. . . . 5 ⊢ (Fun (( I ∘ F) ∖ ( ∼ I ∘ F)) ↔ (( I ∘ F) ∖ ( ∼ I ∘ F)) Fn dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
3	1, 2	mpbi 199	. . . 4 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) Fn dom (( I ∘ F) ∖ ( ∼ I ∘ F))
4		0ex 4111	. . . . 5 ⊢ ∅ ∈ V
5		fnconstg 5253	. . . . 5 ⊢ (∅ ∈ V → ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) Fn ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
6	4, 5	ax-mp 5	. . . 4 ⊢ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) Fn ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))
7	3, 6	pm3.2i 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 4074	. . 3 ⊢ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∩ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) = ∅
9		fnun 5190	. . 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))))
10	7, 8, 9	mp2an 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 5769	. . 3 ⊢ FullFun F = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}))
12		uncompl 4075	. . . 4 ⊢ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) = V
13	12	eqcomi 2357	. . 3 ⊢ V = (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
14		fneq1 5174	. . . 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 5175	. . . 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)))))
16	14, 15	sylan9bb 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)))))
17	11, 13, 16	mp2an 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))))
18	10, 17	mpbir 200	1 ⊢ FullFun F Fn V

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738   ∘ ccom 4722   I cid 4764   × cxp 4771  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   FullFun cfullfun 5768

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-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-fullfun 5769

This theorem is referenced by:  brfullfung  5866