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Theorem mpt2mptx 6393
Description: Express a two-argument function as a one-argument function, or vice-versa. In this version (x) 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:   , , ,   , ,   , ,   ,

Proof of Theorem mpt2mptx
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4512 . 2
2 df-mpt2 6301 . . 3
3 eliunxp 5145 . . . . . . 7
43anbi1i 695 . . . . . 6
5 19.41vv 1772 . . . . . 6
6 anass 649 . . . . . . . 8
7 mpt2mpt.1 . . . . . . . . . . 11
87eqeq2d 2471 . . . . . . . . . 10
98anbi2d 703 . . . . . . . . 9
109pm5.32i 637 . . . . . . . 8
116, 10bitri 249 . . . . . . 7
12112exbii 1668 . . . . . 6
134, 5, 123bitr2i 273 . . . . 5
1413opabbii 4516 . . . 4
15 dfoprab2 6343 . . . 4
1614, 15eqtr4i 2489 . . 3
172, 16eqtr4i 2489 . 2
181, 17eqtr4i 2489 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {csn 4029  <.cop 4035  U_ciun 4330  {copab 4509  e.cmpt 4510  X.cxp 5002  {coprab 6297  e.cmpt2 6298
This theorem is referenced by:  mpt2mpt  6394  mpt2mptsx  6863  dmmpt2ssx  6865  fmpt2x  6866  gsumcom2  17003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iun 4332  df-opab 4511  df-mpt 4512  df-xp 5010  df-rel 5011  df-oprab 6300  df-mpt2 6301
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