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Theorem fvmptex 5721
 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 5339.) 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 3139 . . . 4
2 fvmptex.1 . . . . 5
3 nfcv 2489 . . . . . 6
4 nfcsb1v 3168 . . . . . 6
5 csbeq1a 3144 . . . . . 6
63, 4, 5cbvmpt 5676 . . . . 5
72, 6eqtri 2373 . . . 4
81, 7fvmpti 5699 . . 3
91fveq2d 5332 . . . 4
10 fvmptex.2 . . . . 5
11 nfcv 2489 . . . . . 6
12 nfcv 2489 . . . . . . 7
1312, 4nffv 5334 . . . . . 6
145fveq2d 5332 . . . . . 6
1511, 13, 14cbvmpt 5676 . . . . 5
1610, 15eqtri 2373 . . . 4
17 fvex 5339 . . . 4
189, 16, 17fvmpt 5700 . . 3
198, 18eqtr4d 2388 . 2
202dmmptss 5685 . . . . . 6
2120sseli 3269 . . . . 5
2221con3i 127 . . . 4
23 ndmfv 5349 . . . 4
2422, 23syl 15 . . 3
25 fvex 5339 . . . . . 6
2625, 10dmmpti 5691 . . . . 5
2726eleq2i 2417 . . . 4
28 ndmfv 5349 . . . 4
2927, 28sylnbir 298 . . 3
3024, 29eqtr4d 2388 . 2
3119, 30pm2.61i 156 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1642   wcel 1710  csb 3136  c0 3550   cid 4763   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:  fvmptnf  5723
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