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Theorem fvmptex 5768
 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 5696.) 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 3211 . . . 4
2 fvmptex.1 . . . . 5
3 nfcv 2537 . . . . . 6
4 nfcsb1v 3240 . . . . . 6
5 csbeq1a 3216 . . . . . 6
63, 4, 5cbvmpt 4254 . . . . 5
72, 6eqtri 2421 . . . 4
81, 7fvmpti 5758 . . 3
91fveq2d 5686 . . . 4
10 fvmptex.2 . . . . 5
11 nfcv 2537 . . . . . 6
12 nfcv 2537 . . . . . . 7
1312, 4nffv 5689 . . . . . 6
145fveq2d 5686 . . . . . 6
1511, 13, 14cbvmpt 4254 . . . . 5
1610, 15eqtri 2421 . . . 4
17 fvex 5696 . . . 4
189, 16, 17fvmpt 5759 . . 3
198, 18eqtr4d 2436 . 2
202dmmptss 5320 . . . . . 6
2120sseli 3301 . . . . 5
2221con3i 129 . . . 4
23 ndmfv 5709 . . . 4
2422, 23syl 16 . . 3
25 fvex 5696 . . . . . 6
2625, 10dmmpti 5528 . . . . 5
2726eleq2i 2465 . . . 4
28 ndmfv 5709 . . . 4
2927, 28sylnbir 299 . . 3
3024, 29eqtr4d 2436 . 2
3119, 30pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1649   wcel 1721  csb 3208  c0 3585   cmpt 4221   cid 4448   cdm 4832  cfv 5408 This theorem is referenced by:  fvmptnf  5775  sumeq2ii  12428  prodeq2ii  25161 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2382  ax-sep 4285  ax-nul 4293  ax-pow 4332  ax-pr 4358 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2256  df-mo 2257  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-ne 2566  df-ral 2668  df-rex 2669  df-rab 2672  df-v 2915  df-sbc 3119  df-csb 3209  df-dif 3280  df-un 3282  df-in 3284  df-ss 3291  df-nul 3586  df-if 3697  df-sn 3777  df-pr 3778  df-op 3780  df-uni 3972  df-br 4168  df-opab 4222  df-mpt 4223  df-id 4453  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5372  df-fun 5410  df-fn 5411  df-fv 5416
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