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Theorem mpt2mptx 6156
 Description: Express a two-argument function as a one-argument function, or vice-versa. In this version is not assumed to be constant w.r.t . (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpt2mpt.1
Assertion
Ref Expression
mpt2mptx
Distinct variable groups:   ,,,   ,,   ,,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem mpt2mptx
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4260 . 2
2 df-mpt2 6078 . . 3
3 eliunxp 5004 . . . . . . 7
43anbi1i 677 . . . . . 6
5 19.41vv 1925 . . . . . 6
6 anass 631 . . . . . . . 8
7 mpt2mpt.1 . . . . . . . . . . 11
87eqeq2d 2446 . . . . . . . . . 10
98anbi2d 685 . . . . . . . . 9
109pm5.32i 619 . . . . . . . 8
116, 10bitri 241 . . . . . . 7
12112exbii 1593 . . . . . 6
134, 5, 123bitr2i 265 . . . . 5
1413opabbii 4264 . . . 4
15 dfoprab2 6113 . . . 4
1614, 15eqtr4i 2458 . . 3
172, 16eqtr4i 2458 . 2
181, 17eqtr4i 2458 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wex 1550   wceq 1652   wcel 1725  csn 3806  cop 3809  ciun 4085  copab 4257   cmpt 4258   cxp 4868  coprab 6074   cmpt2 6075 This theorem is referenced by:  mpt2mpt  6157  mpt2mptsx  6406  dmmpt2ssx  6408  fmpt2x  6409  gsumcom2  15541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-iun 4087  df-opab 4259  df-mpt 4260  df-xp 4876  df-rel 4877  df-oprab 6077  df-mpt2 6078
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