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Theorem fvmptex 5975
 Description: Express a function whose value may not always be a set in terms of another function for which sethood is guaranteed. (Note that is just shorthand for , and it is always a set by fvex 5889.) Note also that these functions are not the same; wherever is not a set, is not in the domain of (so it evaluates to the empty set), but is in the domain of , and is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
fvmptex.1
fvmptex.2
Assertion
Ref Expression
fvmptex
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem fvmptex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . . 4
2 fvmptex.1 . . . . 5
3 nfcv 2612 . . . . . 6
4 nfcsb1v 3365 . . . . . 6
5 csbeq1a 3358 . . . . . 6
63, 4, 5cbvmpt 4487 . . . . 5
72, 6eqtri 2493 . . . 4
81, 7fvmpti 5962 . . 3
91fveq2d 5883 . . . 4
10 fvmptex.2 . . . . 5
11 nfcv 2612 . . . . . 6
12 nfcv 2612 . . . . . . 7
1312, 4nffv 5886 . . . . . 6
145fveq2d 5883 . . . . . 6
1511, 13, 14cbvmpt 4487 . . . . 5
1610, 15eqtri 2493 . . . 4
17 fvex 5889 . . . 4
189, 16, 17fvmpt 5963 . . 3
198, 18eqtr4d 2508 . 2
202dmmptss 5338 . . . . . 6
2120sseli 3414 . . . . 5
2221con3i 142 . . . 4
23 ndmfv 5903 . . . 4
2422, 23syl 17 . . 3
25 fvex 5889 . . . . . 6
2625, 10dmmpti 5717 . . . . 5
2726eleq2i 2541 . . . 4
28 ndmfv 5903 . . . 4
2927, 28sylnbir 314 . . 3
3024, 29eqtr4d 2508 . 2
3119, 30pm2.61i 169 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1452   wcel 1904  csb 3349  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:  fvmptnf  5982  sumeq2ii  13836  prodeq2ii  14044
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