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

Theorem fvmptnf 5723
 Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5724 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvmptf.1 xA
fvmptf.2 xC
fvmptf.3 (x = AB = C)
fvmptf.4 F = (x D B)
Assertion
Ref Expression
fvmptnf C V → (FA) = )
Distinct variable group:   x,D
Allowed substitution hints:   A(x)   B(x)   C(x)   F(x)

Proof of Theorem fvmptnf
StepHypRef Expression
1 fvmptf.4 . . . . 5 F = (x D B)
21dmmptss 5685 . . . 4 dom F D
32sseli 3269 . . 3 (A dom FA D)
4 eqid 2353 . . . . . . 7 (x D ( I ‘B)) = (x D ( I ‘B))
51, 4fvmptex 5721 . . . . . 6 (FA) = ((x D ( I ‘B)) ‘A)
6 fvex 5339 . . . . . . 7 ( I ‘C) V
7 fvmptf.1 . . . . . . . 8 xA
8 nfcv 2489 . . . . . . . . 9 x I
9 fvmptf.2 . . . . . . . . 9 xC
108, 9nffv 5334 . . . . . . . 8 x( I ‘C)
11 fvmptf.3 . . . . . . . . 9 (x = AB = C)
1211fveq2d 5332 . . . . . . . 8 (x = A → ( I ‘B) = ( I ‘C))
137, 10, 12, 4fvmptf 5722 . . . . . . 7 ((A D ( I ‘C) V) → ((x D ( I ‘B)) ‘A) = ( I ‘C))
146, 13mpan2 652 . . . . . 6 (A D → ((x D ( I ‘B)) ‘A) = ( I ‘C))
155, 14syl5eq 2397 . . . . 5 (A D → (FA) = ( I ‘C))
16 fvprc 5325 . . . . 5 C V → ( I ‘C) = )
1715, 16sylan9eq 2405 . . . 4 ((A D ¬ C V) → (FA) = )
1817expcom 424 . . 3 C V → (A D → (FA) = ))
193, 18syl5 28 . 2 C V → (A dom F → (FA) = ))
20 ndmfv 5349 . 2 A dom F → (FA) = )
2119, 20pm2.61d1 151 1 C V → (FA) = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  ∅c0 3550   I cid 4763  dom cdm 4772   ‘cfv 4781   ↦ cmpt 5651 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-csb 3137  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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652 This theorem is referenced by:  fvmptn  5724
 Copyright terms: Public domain W3C validator