New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  fvfullfun GIF version

Theorem fvfullfun 5864
 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 FA) = (FA)

Proof of Theorem fvfullfun
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5328 . . . 4 (x = A → ( FullFun Fx) = ( FullFun FA))
2 fveq2 5328 . . . 4 (x = A → (Fx) = (FA))
31, 2eqeq12d 2367 . . 3 (x = A → (( FullFun Fx) = (Fx) ↔ ( FullFun FA) = (FA)))
4 df-fullfun 5768 . . . . 5 FullFun F = ((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {}))
54fveq1i 5329 . . . 4 ( FullFun Fx) = (((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) ‘x)
6 incompl 4073 . . . . . . 7 (dom (( I F) ( ∼ I F)) ∩ ∼ dom (( I F) ( ∼ I F))) =
7 fnfullfunlem2 5857 . . . . . . . . 9 Fun (( I F) ( ∼ I F))
8 funfn 5136 . . . . . . . . 9 (Fun (( I F) ( ∼ I F)) ↔ (( I F) ( ∼ I F)) Fn dom (( I F) ( ∼ I F)))
97, 8mpbi 199 . . . . . . . 8 (( I F) ( ∼ I F)) Fn dom (( I F) ( ∼ I F))
10 0ex 4110 . . . . . . . . 9 V
11 fnconstg 5252 . . . . . . . . 9 ( V → ( ∼ dom (( I F) ( ∼ I F)) × {}) Fn ∼ dom (( I F) ( ∼ I F)))
1210, 11ax-mp 8 . . . . . . . 8 ( ∼ dom (( I F) ( ∼ I F)) × {}) Fn ∼ dom (( I F) ( ∼ I F))
13 fvun1 5379 . . . . . . . 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))
149, 12, 13mp3an12 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))
156, 14mpan 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 5863 . . . . . 6 (x dom (( I F) ( ∼ I F)) → ((( I F) ( ∼ I F)) ‘x) = (Fx))
1715, 16eqtrd 2385 . . . . 5 (x dom (( I F) ( ∼ I F)) → (((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) ‘x) = (Fx))
18 vex 2862 . . . . . . . 8 x V
1918elcompl 3225 . . . . . . 7 (x ∼ dom (( I F) ( ∼ I F)) ↔ ¬ x dom (( I F) ( ∼ I F)))
20 fvun2 5380 . . . . . . . . 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))
219, 12, 20mp3an12 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))
226, 21mpan 651 . . . . . . 7 (x ∼ dom (( I F) ( ∼ I F)) → (((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) ‘x) = (( ∼ dom (( I F) ( ∼ I F)) × {}) ‘x))
2319, 22sylbir 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 5861 . . . . . . . . 9 dom (( I F) ( ∼ I F)) = {x ∃!y xFy}
2524abeq2i 2460 . . . . . . . 8 (x dom (( I F) ( ∼ I F)) ↔ ∃!y xFy)
26 tz6.12-2 5346 . . . . . . . 8 ∃!y xFy → (Fx) = )
2725, 26sylnbi 297 . . . . . . 7 x dom (( I F) ( ∼ I F)) → (Fx) = )
2810fvconst2 5453 . . . . . . . 8 (x ∼ dom (( I F) ( ∼ I F)) → (( ∼ dom (( I F) ( ∼ I F)) × {}) ‘x) = )
2919, 28sylbir 204 . . . . . . 7 x dom (( I F) ( ∼ I F)) → (( ∼ dom (( I F) ( ∼ I F)) × {}) ‘x) = )
3027, 29eqtr4d 2388 . . . . . 6 x dom (( I F) ( ∼ I F)) → (Fx) = (( ∼ dom (( I F) ( ∼ I F)) × {}) ‘x))
3123, 30eqtr4d 2388 . . . . 5 x dom (( I F) ( ∼ I F)) → (((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) ‘x) = (Fx))
3217, 31pm2.61i 156 . . . 4 (((( I F) ( ∼ I F)) ∪ ( ∼ dom (( I F) ( ∼ I F)) × {})) ‘x) = (Fx)
335, 32eqtri 2373 . . 3 ( FullFun Fx) = (Fx)
343, 33vtoclg 2914 . 2 (A V → ( FullFun FA) = (FA))
35 fvprc 5325 . . 3 A V → ( FullFun FA) = )
36 fvprc 5325 . . 3 A V → (FA) = )
3735, 36eqtr4d 2388 . 2 A V → ( FullFun FA) = (FA))
3834, 37pm2.61i 156 1 ( FullFun FA) = (FA)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737   class class class wbr 4639   ∘ ccom 4721   I cid 4763   × cxp 4770  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   ‘cfv 4781   FullFun cfullfun 5767 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-fullfun 5768 This theorem is referenced by:  brfullfung  5865
 Copyright terms: Public domain W3C validator