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

Theorem pmex 6005
 Description: The class of all partial functions from one set to another is a set. (Contributed by set.mm contributors, 15-Nov-2007.)
Assertion
Ref Expression
pmex ((A C B D) → {f (Fun f f (A × B))} V)
Distinct variable groups:   A,f   B,f
Allowed substitution hints:   C(f)   D(f)

Proof of Theorem pmex
StepHypRef Expression
1 df-funs 5760 . . . 4 Funs = {f Fun f}
2 df-pw 3724 . . . 4 (A × B) = {f f (A × B)}
31, 2ineq12i 3455 . . 3 ( Funs(A × B)) = ({f Fun f} ∩ {f f (A × B)})
4 inab 3522 . . 3 ({f Fun f} ∩ {f f (A × B)}) = {f (Fun f f (A × B))}
53, 4eqtr2i 2374 . 2 {f (Fun f f (A × B))} = ( Funs(A × B))
6 xpexg 5114 . . 3 ((A C B D) → (A × B) V)
7 pwexg 4328 . . 3 ((A × B) V → (A × B) V)
8 funsex 5828 . . . 4 Funs V
9 inexg 4100 . . . 4 (( Funs V (A × B) V) → ( Funs(A × B)) V)
108, 9mpan 651 . . 3 ((A × B) V → ( Funs(A × B)) V)
116, 7, 103syl 18 . 2 ((A C B D) → ( Funs(A × B)) V)
125, 11syl5eqel 2437 1 ((A C B D) → {f (Fun f f (A × B))} V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  {cab 2339  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ℘cpw 3722   × cxp 4770  Fun wfun 4775   Funs cfuns 5759 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-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-fun 4789  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-funs 5760 This theorem is referenced by:  pmvalg  6010
 Copyright terms: Public domain W3C validator