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Theorem fvmptnf 5982
 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 5983 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
fvmptf.2
fvmptf.3
fvmptf.4
Assertion
Ref Expression
fvmptnf
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem fvmptnf
StepHypRef Expression
1 fvmptf.4 . . . . 5
21dmmptss 5338 . . . 4
32sseli 3414 . . 3
4 eqid 2471 . . . . . . 7
51, 4fvmptex 5975 . . . . . 6
6 fvex 5889 . . . . . . 7
7 fvmptf.1 . . . . . . . 8
8 nfcv 2612 . . . . . . . . 9
9 fvmptf.2 . . . . . . . . 9
108, 9nffv 5886 . . . . . . . 8
11 fvmptf.3 . . . . . . . . 9
1211fveq2d 5883 . . . . . . . 8
137, 10, 12, 4fvmptf 5981 . . . . . . 7
146, 13mpan2 685 . . . . . 6
155, 14syl5eq 2517 . . . . 5
16 fvprc 5873 . . . . 5
1715, 16sylan9eq 2525 . . . 4
1817expcom 442 . . 3
193, 18syl5 32 . 2
20 ndmfv 5903 . 2
2119, 20pm2.61d1 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1452   wcel 1904  wnfc 2599  cvv 3031  c0 3722   cmpt 4454   cid 4749   cdm 4839  cfv 5589 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597 This theorem is referenced by:  fvmptn  5983  rdgsucmptnf  7165  frsucmptn  7174
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