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.

Step	Hyp	Ref	Expression
1		df-mpt 4512	. 2
2		df-mpt2 6301	. . 3
3		eliunxp 5145	. . . . . . 7
4	3	anbi1i 695	. . . . . 6
5		19.41vv 1772	. . . . . 6
6		anass 649	. . . . . . . 8
7		mpt2mpt.1	. . . . . . . . . . 11
8	7	eqeq2d 2471	. . . . . . . . . 10
9	8	anbi2d 703	. . . . . . . . 9
10	9	pm5.32i 637	. . . . . . . 8
11	6, 10	bitri 249	. . . . . . 7
12	11	2exbii 1668	. . . . . 6
13	4, 5, 12	3bitr2i 273	. . . . 5
14	13	opabbii 4516	. . . 4
15		dfoprab2 6343	. . . 4
16	14, 15	eqtr4i 2489	. . 3
17	2, 16	eqtr4i 2489	. 2
18	1, 17	eqtr4i 2489	1

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