Theorem fvfullfun 5865

Description: The value of the full function definition agrees with the function value everywhere. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
fvfullfun	⊢ ( FullFun F ‘A) = (F ‘A)

Proof of Theorem fvfullfun

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5329	. . . 4 ⊢ (x = A → ( FullFun F ‘x) = ( FullFun F ‘A))
2		fveq2 5329	. . . 4 ⊢ (x = A → (F ‘x) = (F ‘A))
3	1, 2	eqeq12d 2367	. . 3 ⊢ (x = A → (( FullFun F ‘x) = (F ‘x) ↔ ( FullFun F ‘A) = (F ‘A)))
4		df-fullfun 5769	. . . . 5 ⊢ FullFun F = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}))
5	4	fveq1i 5330	. . . 4 ⊢ ( FullFun F ‘x) = (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x)
6		incompl 4074	. . . . . . 7 ⊢ (dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∩ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) = ∅
7		fnfullfunlem2 5858	. . . . . . . . 9 ⊢ Fun (( I ∘ F) ∖ ( ∼ I ∘ F))
8		funfn 5137	. . . . . . . . 9 ⊢ (Fun (( I ∘ F) ∖ ( ∼ I ∘ F)) ↔ (( I ∘ F) ∖ ( ∼ I ∘ F)) Fn dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
9	7, 8	mpbi 199	. . . . . . . 8 ⊢ (( I ∘ F) ∖ ( ∼ I ∘ F)) Fn dom (( I ∘ F) ∖ ( ∼ I ∘ F))
10		0ex 4111	. . . . . . . . 9 ⊢ ∅ ∈ V
11		fnconstg 5253	. . . . . . . . 9 ⊢ (∅ ∈ V → ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) Fn ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) Fn ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))
13		fvun1 5380	. . . . . . . 8 ⊢ (((( 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))) = ∅ ∧ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘x))
14	9, 12, 13	mp3an12 1267	. . . . . . 7 ⊢ (((dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∩ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) = ∅ ∧ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘x))
15	6, 14	mpan 651	. . . . . 6 ⊢ (x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘x))
16		fvfullfunlem3 5864	. . . . . 6 ⊢ (x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → ((( I ∘ F) ∖ ( ∼ I ∘ F)) ‘x) = (F ‘x))
17	15, 16	eqtrd 2385	. . . . 5 ⊢ (x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (F ‘x))
18		vex 2863	. . . . . . . 8 ⊢ x ∈ V
19	18	elcompl 3226	. . . . . . 7 ⊢ (x ∈ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ↔ ¬ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))
20		fvun2 5381	. . . . . . . . 9 ⊢ (((( 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))) = ∅ ∧ x ∈ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)))) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x))
21	9, 12, 20	mp3an12 1267	. . . . . . . 8 ⊢ (((dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ∩ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) = ∅ ∧ x ∈ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F))) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x))
22	6, 21	mpan 651	. . . . . . 7 ⊢ (x ∈ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x))
23	19, 22	sylbir 204	. . . . . 6 ⊢ (¬ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x))
24		fvfullfunlem1 5862	. . . . . . . . 9 ⊢ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) = {x ∣ ∃!y xFy}
25	24	abeq2i 2461	. . . . . . . 8 ⊢ (x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) ↔ ∃!y xFy)
26		tz6.12-2 5347	. . . . . . . 8 ⊢ (¬ ∃!y xFy → (F ‘x) = ∅)
27	25, 26	sylnbi 297	. . . . . . 7 ⊢ (¬ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (F ‘x) = ∅)
28	10	fvconst2 5454	. . . . . . . 8 ⊢ (x ∈ ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x) = ∅)
29	19, 28	sylbir 204	. . . . . . 7 ⊢ (¬ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x) = ∅)
30	27, 29	eqtr4d 2388	. . . . . 6 ⊢ (¬ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (F ‘x) = (( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅}) ‘x))
31	23, 30	eqtr4d 2388	. . . . 5 ⊢ (¬ x ∈ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) → (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (F ‘x))
32	17, 31	pm2.61i 156	. . . 4 ⊢ (((( I ∘ F) ∖ ( ∼ I ∘ F)) ∪ ( ∼ dom (( I ∘ F) ∖ ( ∼ I ∘ F)) × {∅})) ‘x) = (F ‘x)
33	5, 32	eqtri 2373	. . 3 ⊢ ( FullFun F ‘x) = (F ‘x)
34	3, 33	vtoclg 2915	. 2 ⊢ (A ∈ V → ( FullFun F ‘A) = (F ‘A))
35		fvprc 5326	. . 3 ⊢ (¬ A ∈ V → ( FullFun F ‘A) = ∅)
36		fvprc 5326	. . 3 ⊢ (¬ A ∈ V → (F ‘A) = ∅)
37	35, 36	eqtr4d 2388	. 2 ⊢ (¬ A ∈ V → ( FullFun F ‘A) = (F ‘A))
38	34, 37	pm2.61i 156	1 ⊢ ( FullFun F ‘A) = (F ‘A)

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738   class class class wbr 4640   ∘ ccom 4722   I cid 4764   × cxp 4771  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   ‘cfv 4782   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-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-fullfun 5769

This theorem is referenced by:  brfullfung  5866