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Theorem mpt2mptsx 6863
 Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
mpt2mptsx
Distinct variable groups:   ,,,   ,,   ,

Proof of Theorem mpt2mptsx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . 6
2 vex 3112 . . . . . 6
31, 2op1std 6810 . . . . 5
43csbeq1d 3441 . . . 4
51, 2op2ndd 6811 . . . . . 6
65csbeq1d 3441 . . . . 5
76csbeq2dv 3835 . . . 4
84, 7eqtrd 2498 . . 3
98mpt2mptx 6393 . 2
10 nfcv 2619 . . . 4
11 nfcv 2619 . . . . 5
12 nfcsb1v 3450 . . . . 5
1311, 12nfxp 5031 . . . 4
14 sneq 4039 . . . . 5
15 csbeq1a 3443 . . . . 5
1614, 15xpeq12d 5029 . . . 4
1710, 13, 16cbviun 4367 . . 3
18 mpteq1 4532 . . 3
1917, 18ax-mp 5 . 2
20 nfcv 2619 . . 3
21 nfcv 2619 . . 3
22 nfcv 2619 . . 3
23 nfcsb1v 3450 . . 3
24 nfcv 2619 . . . 4
25 nfcsb1v 3450 . . . 4
2624, 25nfcsb 3452 . . 3
27 csbeq1a 3443 . . . 4
28 csbeq1a 3443 . . . 4
2927, 28sylan9eqr 2520 . . 3
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 6375 . 2
319, 19, 303eqtr4ri 2497 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  [_csb 3434  {csn 4029  <.cop 4035  U_ciun 4330  e.cmpt 4510  X.cxp 5002  cfv 5593  e.`cmpt2 6298   c1st 6798   c2nd 6799 This theorem is referenced by:  mpt2mpts  6864  ovmptss  6881 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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592 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-eu 2286  df-mo 2287  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-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801
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